if the input is: DelSeqIntV(x^(4*n)*(1-x)^(4*n)/(x^2+1),x,n,0,1,Pi,A,100): the output is: Better and Better Rational Approximations to, Pi By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / (4 n) (4 n) | x (1 - x) A(n) = | ------------------- dx | 2 / x + 1 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: Pi A(0) = ----, A(1) = 22/7 - Pi 4 Note that these are linear combinations of, Pi, and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients 3 2 -4 (4 n + 1) (4 n + 3) (820 n + 3993 n + 6428 n + 3420) (n + 1) (2 n + 1) 7 6 5 4 A(n) + (26843520 n + 211258528 n + 688917624 n + 1202271190 n 3 2 + 1207256235 n + 693651577 n + 209656416 n + 25472340) A(n + 1) + 2 3 2 (8 n + 9) (8 n + 11) (8 n + 13) (8 n + 15) (820 n + 1533 n + 902 n + 165) A(n + 2) = 0 and in Maple input form : -4*(4*n+1)*(4*n+3)*(820*n^3+3993*n^2+6428*n+3420)*(n+1)*(2*n+1)*A(n)+(26843520* n^7+211258528*n^6+688917624*n^5+1202271190*n^4+1207256235*n^3+693651577*n^2+ 209656416*n+25472340)*A(n+1)+2*(8*n+9)*(8*n+11)*(8*n+13)*(8*n+15)*(820*n^3+1533 *n^2+902*n+165)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , Pi, and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , Pi, yields the following sequence of terrific approximations to, Pi `The `, 1, `-th apprx. is`, 0 the error is, -3.141592654 `The `, 2, `-th apprx. is`, 22/7 the error is, 0.001264489267 `The `, 3, `-th apprx. is`, 47171/15015 -6 the error is, -0.9119980516 10 `The `, 4, `-th apprx. is`, 431302721/137287920 -9 the error is, 0.7384244226 10 `The `, 5, `-th apprx. is`, 741269838109/235953517800 -12 the error is, -0.6294154514 10 `The `, 6, `-th apprx. is`, 26856502742629699/8548690331301120 -15 the error is, 0.5523906097 10 `The `, 7, `-th apprx. is`, 211938730834003723543/67462193289708771840 -18 the error is, -0.4940165130 10 `The `, 8, `-th apprx. is`, 506119433541064524255449/161102819285860855603200 -21 the error is, 0.4476749493 10 `The `, 9, `-th apprx. is`, 19809071774292917047896724979/ 6305423381881718760060595200 -24 the error is, -0.4096524271 10 `The `, 10, `-th apprx. is`, 23285066731814401580687501596643/ 7411866941185812791748757094400 -27 the error is, 0.3776793082 10 `The `, 11, `-th apprx. is`, 89293478252053341114758995682016773/ 28422996899365886608045972478361600 -30 the error is, -0.3502777794 10 `The `, 12, `-th apprx. is`, 77997614500552946262295151821844788921/ 24827411794278189209115835981312819200 -33 the error is, 0.3264374199 10 `The `, 13, `-th apprx. is`, 53385652953509417766719017367204093188361/ 16993181115478931076058403260828562227200 -36 the error is, -0.3054399636 10 `The `, 14, `-th apprx. is`, 2504492442679095809835466877865587690003305358231/ 797204704377346869713059903864272972590717337600 -39 the error is, 0.2867583520 10 `The `, 15, `-th apprx. is`, 27866240172453844886325941887505217163994611104897858617/ 8870099737663958700556100446465640958076561638241075200 -42 the error is, -0.2699954791 10 `The `, 16, `-th apprx. is`, 3737271881299351053666441876167156192604573934552162661/ 1189610587174277674549768534026398549974217301531033600 -45 the error is, 0.2548453920 10 `The `, 17, `-th apprx. is`, 2199496269931868203800149756151730543362676840844586435379483/ 700121407343685092214243286289756273641462048387187186073600 -48 the error is, -0.2410678137 10 `The `, 18, `-th apprx. is`, 132570645054700093606473440370784391628703685360477563188116477/ 42198546938673298440864239484207883524155412032193942257664000 -51 the error is, 0.2284708900 10 `The `, 19, `-th apprx. is`, 29832337484492518217048916227659410964608663939042391473810429697/ 9495927949285245581337958144499514323433226651874145194402119680 -54 the error is, -0.2168991928 10 `The `, 20, `-th apprx. is`, 842755908791452603343975605728390570004752202303444739958124097773741955147/ 268257537408124350364026727600009545561127583382712382710348191778209792000 -57 the error is, 0.2062251819 10 `The `, 21, `-th apprx. is`, 66156338840129029362502085049678659745373047880820412086708394806039520515257/ 21058216686537761503576098116600749326548515295542922042762333054589468672000 -60 the error is, -0.1963430041 10 `The `, 22, `-th apprx. is`, 19208978128716291288284472076131229833907197187880\ 665999064419495161046612530817093/611440764185860699614767964620788263912508943\ 8506388063211023659509119218352128000 -63 the error is, 0.1871639069 10 `The `, 23, `-th apprx. is`, 19416176095048511958360349527738937481553836618221\ 8507109644401986701079185812018101/61803608029393290491398343929534957552325061\ 335734232602973448135892175829545779200 -66 the error is, -0.1786127913 10 `The `, 24, `-th apprx. is`, 11376531846179208486779499075860814093243136394380\ 595776565666024223647991552013566339271/362126255712357381677708976586430835634\ 5692454910121468879503589693158929508344004608000 -69 the error is, 0.1706255832 10 `The `, 25, `-th apprx. is`, 33072589649906601294503039619501915430104748241262\ 345867203946584680606098687119416797781273/105273322472649836299500690462668287\ 18284283446809533417225181736459261497727838593220608000 -72 the error is, -0.1631472016 10 `The `, 26, `-th apprx. is`, 25422588845948287917840198990310955266893220891255\ 517493881289524342496083962053135792429098585663/809226136205110576524828530168\ 5603166696861085875287087233705766880133617989065376859671442227200 -75 the error is, 0.1561299707 10 `The `, 27, `-th apprx. is`, 46264531052986974409042524288673990665877753333636\ 166970847031386641299179820533992829502611931261/147264576138227330142356182967\ 61187744799693147268540465055933017205288205709830685816699381350400 -78 the error is, -0.1495323636 10 `The `, 28, `-th apprx. is`, 15118337974655890138153621692618786255874458647572\ 745177428650890392802385633565461429689136646412227/481231644000079676800114183\ 5184294518755566891467916410309369717790009488585371350904855292228403200 -81 the error is, 0.1433180002 10 `The `, 29, `-th apprx. is`, 20699286486013274273524398451336424697562876710582\ 25941488213320736310619376143943839757508672264691186711/6588787525448561692109\ 2999818717157284154628991393473472566937480840531545798142953921954110519136747\ 5200 -84 the error is, -0.1374548386 10 `The `, 30, `-th apprx. is`, 19454279049995898901015987988900450280165671242435\ 25244824182818377188279480704038637893031935822360758977292080733/6192489350191\ 8978905317927565140841686807958225781693321962003768082922992321665562340946924\ 7302624426776854528000 -87 the error is, 0.1319145190 10 `The `, 31, `-th apprx. is`, 23173271389457146996810210120596778069552636031604\ 1048788287205576407451806292974972467047521040580190265939101075713/73762813784\ 8419717283516456111229529628507972132428910378630610445611493619982158325150868\ 74811852933591278498611200 -90 the error is, -0.1266718253 10 `The `, 32, `-th apprx. is`, 32763915975173811567223395749174431286604820330550\ 65708467996703909632825271907608810720762683053048665415846495441582581/1042907\ 8364992856909432544667471303909574264715323034887539998657553685704461721075839\ 86641627339851343735889599030886400 -93 the error is, 0.1217042415 10 `The `, 33, `-th apprx. is`, 14744914045249217409563438284635300992572088849462\ 541609644118447684565944013718894666965951091297717732625917014942453716139/469\ 3451911533054889370367338260552547336490092139263774838264473976963188770821966\ 509512382626726442713299535344326148096000 -96 the error is, -0.1169915831 10 `The `, 34, `-th apprx. is`, 18094724270886851248654567618211125186068698914932\ 6098144672791602373446213325394393769739910393090031441224715144284011192972595\ 9/57597296231930682648952283462875507695118652244931173275250951999650844848514\ 4742197277869306776632264505425508690335184715776000 -99 the error is, 0.1125156889 10 `The `, 35, `-th apprx. is`, 11456546857118892497691297840820238806642047096624\ 7064907415612352583724883811285691182990423580163702720142801163265233956621347\ 5573/36467321261487793997601535262224447355986947724318981285752268325170415987\ 2045205282758919254541505428468725430181426036706639872000 -102 the error is, -0.1082601616 10 `The `, 36, `-th apprx. is`, 23997481388844424985289836072762754344375521101355\ 5059862043452073563515999052998410397597193481954205079096238042093742080440181\ 303577497/763863556958070380847395439452410750492754904146894676990239396493131\ 71834113902960821522439577785262595061187538615894965340914843648000 -105 the error is, 0.1042101485 10 `The `, 37, `-th apprx. is`, 38334433522726510759306283525375225056461436428102\ 5003229712758735730881540309331114308861390381911116961885356141120111561911232\ 48109114121841/1220222917153916532913436921467584189957780241011090590895571133\ 6628981423701850335790621699407216015729407672254465241804476284875302764544000 -108 the error is, -0.1003521558 10 `The `, 38, `-th apprx. is`, 82346996167252530122317311653792662802349849946953\ 7047770337561775286582393238608924167522668085624424371918731695423872069637824\ 319508091579/262118629775751995211437007297193775745344922693137328508794761264\ 213674109249005725877301871080393456839150170381790884403688867245326336000 -112 the error is, 0.9667388860 10 `The `, 39, `-th apprx. is`, 28834831775394475155689761741074359292575367196542\ 8225933936451902398489068755237784774543857261907842449224941868306504138154425\ 05610003720392059/9178412020554566087515748357688444107072715126868953788508119\ 2414599574950483142556752336890325084725099209023199132556059815198852608121896\ 96000 -115 the error is, -0.9316411472 10 `The `, 40, `-th apprx. is`, 12229841572596113323658717227448887731376831402910\ 5795817058477350540613214149467972067973243342549201698492646394496552458046338\ 144142727481148733477/389287947901886035152483391830131186050934184598586071264\ 7166399186498160987848911504758550454839991979645325714472535526603053081024541\ 3326553088000 -118 the error is, 0.8981254689 10 `The `, 41, `-th apprx. is`, 48342614467886085771155676200886759726850439375211\ 3289301183614329951431431719943162945135211932740287931442214823883463206901769\ 08109238458110886946384670429/1538793210909969219974696525398497493212590121009\ 5307633202237616558734891392454997644739224625470055554866306804514274588496291\ 793420876440185453754187776000 -121 the error is, -0.8660974071 10 `The `, 42, `-th apprx. is`, 36198949713553101025441370339224005683465609004158\ 2431028726290410267631856071893440413317246695235927603063930460123937259317264\ 515254375961213439475774828533627/115224835632938495191705275821839492291758748\ 2611936635574183552727918068667467030223638073139955197759948389053522028881186\ 60232949135522784108677711358066688000 -124 the error is, 0.8354700607 10 `The `, 43, `-th apprx. is`, 26333741439969398767958447433589331607092563912915\ 1175100018466210546343174417135667641336282760710092388162991169892358753474340\ 7593084085747780804268735892317859/83822902405500311886713068784678839447411309\ 1746265992033087375720749188415234257151701543317198176831962454454320421010269\ 814002333271605308570820273835606016000 -127 the error is, -0.8061632988 10 `The `, 44, `-th apprx. is`, 36597702278138761663659749054194975803320855693272\ 6266129776692587895871075479921088053906246832566126020922014390887731171095577\ 303485087833259728884015495951490781379/116494104467426057957791125401680973407\ 9577972757969220537577652726180395556411060568314756269611383081048635326503189\ 09130661263765842087924152520316443755833982976000 -130 the error is, 0.7781030841 10 `The `, 45, `-th apprx. is`, 14251892867010647815975358364681415032802134694328\ 5189186445762022818735794604371093868038141642349743713671511833064760317588874\ 93672366747409487670936180509317048640306547/4536518396401737427801471204641115\ 9286504404653604356778910542938124650610875041672053530843185396444741100908778\ 01899177460839196039806629518596991716795264234147545088000 -133 the error is, -0.7512208787 10 `The `, 46, `-th apprx. is`, 85894506034777476312663758748355195171687149117665\ 9073368090439877991024344843450780957949156405711391179855728431117194241121937\ 72458409067715316060651852530698746857173978200017/2734107043974293946598827918\ 2019750513154951641397372773288804859286724929987416012501112557811477441467010\ 441885445859629560125137019816769907955239025886465098186871084679168000 -136 the error is, 0.7254531202 10 `The `, 47, `-th apprx. is`, 64740861567674592392597787127597981751999750871510\ 7024822825159358453798572077278110186431820626265524667770231697451546954461112\ 08563032271068653930832736325787251027176592736118833/2060765627704704721304082\ 7114515672367518384879653008266953629959611195139779759680970926648956176308200\ 807972433183475804504639126341888326616207292082817942146591774514958303232000 -139 the error is, -0.7007407592 10 `The `, 48, `-th apprx. is`, 12939647236883687088193300148910634248162846847107\ 3799671536362254706438518219484486055737014783113038289265794355380388971189258\ 9778866287441317766162016365805242236197314885163029182512669/41188176392308478\ 8743115220678869504976632215500964894526213353792915742577994573743939507817010\ 0712343897712092826693560806990654388220965468392636240342122962696491698123530\ 93703041024000 -142 the error is, 0.6770288489 10 `The `, 49, `-th apprx. is`, 30412570821961226791788777035547618775268367224507\ 2207560252651162668758030912981048290035910862570260669607430557706462465850376\ 70746582526896909210398110815340759036898313158602517760558931/9680621956894951\ 2110725997551947313364464058460935170328273238012497199555365865189360471399277\ 4140384882059900029864568014994077751021681807797069136559814674378456355933480\ 2140973498368000 -145 the error is, -0.6542661799 10 `The `, 50, `-th apprx. is`, 14835554628875585129565082727295405968184028953433\ 2011383560370623932961440401893218517578240518687106340984313918251715460214570\ 4418528277033479342383150636263206948373739234475380098609495073591/47223037053\ 9079635865143599964754416761104121312268331061004338047626542198137051667597391\ 9995809405899746788133330656953363603516771635642157740868844865052998712616793\ 30227940930187914379264000 -148 the error is, 0.6324049560 10 `The `, 51, `-th apprx. is`, 10089338191041135977375265695817794829266997568861\ 3671985190200227192808532977318841104706753154139129563669932236804768821112359\ 47553798088566207468357037391726855793782443210841114365811000623261769/3211536\ 0912600764557719144182298676678668862803227125133869641977710281946760087542496\ 6299399088919848707214394017093347535316580385385088641317178792598305156501143\ 5189455460611496440573538074624000 -151 the error is, -0.6114005028 10 `The `, 52, `-th apprx. is`, 78361234640818860732069116221075807756066963012722\ 8544605818287743494124715494578249691545586813158056392191818964252953968100455\ 95190467175449373878449489333214405411499835479252894151966028842704154343/2494\ 3155679740366443439466677333953263777930242876949482920396459825004664839434728\ 7954878261709230127547506387875339111100482090180581614529588912842453403659655\ 58287791450705869331933818729820848128000 -154 the error is, 0.5912110078 10 `The `, 53, `-th apprx. is`, 13237938138995725364106719612043045697226008034171\ 0717828952476392493322677480616686485934799024457636004863535329808906636654535\ 72178806965661240965422412394297607568584756386811059186562619664173768431453/ 4213766582331791035477567294555698930713441215595582052864660888897828505444940\ 1582058638238516139724373297220438679570268735789625941298402379031341276204344\ 33007683791899205115447357776203248648060928000 -157 the error is, -0.5717972865 10 `The `, 54, `-th apprx. is`, 26592558574015168210711017598232383976597529864424\ 1183365800858455908095396269361456278233880291704512838399120497001694767820338\ 1840591453307532805761762988167113707543592228661848728972169614434835659826366\ 89/8464674293030554953647913374432236135845355923016029364193859827936661073673\ 8163583471165570232125622658792955310037153205235297414678082242064156578730604\ 072717223904706872926196133632952538289909671133184000 -160 the error is, 0.5531225725 10 `The `, 55, `-th apprx. is`, 21471009076383673581116147015596815038598022929401\ 7959724103385123023235610250524235542087889840057501695809039216749148366836593\ 1853972476676857370924085848399344215521102314597184442749697710506311429216492\ 2383393/68344344553548236066083495779390836709891126746159622155125503841415797\ 2862473048650518422225092197752388604907370260644122616807012605147894575471160\ 4428126567237136215302724643026626947009909886166787074031616000 -163 the error is, -0.5351523271 10 `The `, 56, `-th apprx. is`, 15019403704473359564850140138933806551290495175258\ 8130229078355817798833457800144353330279007434979397995452610442346638945433272\ 3589039923520591536405995880526045035145848219556039364776720095664687880401121\ 305853778413081/478082468371931906606644975409588027978369145640538582754929372\ 6722779314420328049876984307978440501865025172869804299050183559684601466750720\ 3709671822834198048139722832703563907259289662326836851569726807125278693785600\ 0 -166 the error is, 0.5178540683 10 `The `, 57, `-th apprx. is`, 83169948013521228590357651019345953777771117032995\ 6771143521395341061040272568299356566420003671198416399818830324494513160336745\ 6874308576495275632848202188412974381982906093018371882354336222362185979593297\ 7330941889548201/26473816686095729328342965513305937049302191439844824020054214\ 0117273904536025665761938006054306142790775768947665413059903914617534806221321\ 1405423077189443716915737151860959851364483165051348590655673621944562307668377\ 6000 -169 the error is, -0.5011972153 10 `The `, 58, `-th apprx. is`, 10436557319286227182412739998654695754401881305939\ 5798146638562517557773158395295416202151289194303745387184252874256219782425487\ 8780524590629555635744654751468276121574954611552839787319895326896263672718673\ 6468546963617200981/33220593724526064540579190261587747032696409310715321530143\ 1394690392267281538582154369019300285647128197925610485297366349296526178174304\ 6237683241814240867015252865624274084816105230927114216292623643545860215828522\ 205184000 -172 the error is, 0.4851529475 10 `The `, 59, `-th apprx. is`, 76086776939404894811156462401835471631007170174279\ 9169334375313311241966231699506449822015840732880720184632207572075637166541729\ 9558785292424657165198668598033166373969388353883238427925067414695260402979073\ 597581882172624262523797/242191733076734408040187548860228562130088882090091599\ 6028233416480772163190471963802730028635624606544969827247558233972929407758002\ 7057962854565978614905308860478147650741906704737170732625166660856852156683962\ 64750519699898368000 -175 the error is, -0.4696940766 10 `The `, 60, `-th apprx. is`, 10451508427682846978136211936304685261397056355095\ 5013451192479902374135399433523187343742840791492075057158243110369154209391749\ 5177964141299554053700990571903108860039821885874189744466187798754230969679219\ 9221442033398138478893115686857773/33268184580646560182724493731287888685338196\ 7431100977728288601368742172129274644782854973260947486096834172268013236077596\ 1996646107353275124762708789701501076395836502809005495431014893663483456836346\ 412282432963926617319404656029335552000 -178 the error is, 0.4547949298 10 `The `, 61, `-th apprx. is`, 18221855342724334652861780292239732984673473080333\ 2064654295690009598933014226751134664238264319446400059615630959695802818296018\ 1226777402935453125583367853030462051245085110772838440759370884078524770069717\ 07342328347764363277317829521917/5800196700200080922881645372820447943611049079\ 1110638542567605756579857483410699152284899247286105350652823948598070932943357\ 2247756213954569688191566669221451406277092831591188873646405352263440279759019\ 97644514439864952172775359905792000 -181 the error is, -0.4404312431 10 `The `, 62, `-th apprx. is`, 43343195119126413566769218969201828355233342704394\ 5401827555970920293718880569888968555323087999177436885483997245958110397998961\ 5158208517926593293670954198747625121921233258373801507495776845309427603107656\ 773747473740746863843750809406786186259/137965675052109600595638948020955016129\ 6870607796537215601974522792721648339773941554327612044462274209187517644022251\ 0138506361664487122143078733713200000680188339306107018046914412073551814431580\ 91706731790862195189428981243524134988786171904000 -184 the error is, 0.4265800644 10 `The `, 63, `-th apprx. is`, 60804310867117340175096247182508850635484489336736\ 3406563799947776754902772456615667316181817735988918602207550422186949158335686\ 0121943949434163734835538604528811242466644371032961543290004047024485231739416\ 60106608290025768246887424908276122597779/1935461327159594682641677813665397511\ 9907527669374279224587699734035037980995114437233567357538027903906030604663283\ 5785085903530778947913492904521519462866684927845694244167458142180803255453883\ 035150858659803810810859894225877242936998288687104000 -187 the error is, -0.4132196642 10 `The `, 64, `-th apprx. is`, 24418719383542161651085812406509078372727520592084\ 4980731670552787395440529885269060238975500330048016963837261653308252284631651\ 4912044104761802871924025092993468856680640539513856397569889982788696709611586\ 600807208907253240982856453629838899191694853971/777271978772922858494421129914\ 5180469074287555887897540054142192229747884985329181443701929663955823674723141\ 8858723013678730445623877743108765689900512993838906935177149125279186100728119\ 5346532482780161136556433633494413936200165527324151449552683008000 -190 the error is, 0.4003294538 10 `The `, 65, `-th apprx. is`, 13493963840830583267182591147548465392794508213515\ 9827500108905304144966854102541724201855338108839156122534617953478821147249242\ 8617277012971353075720592139224130211283228133047605555997987147140999299999418\ 699414677850947045804806767326671812638315803/429526209434296978110300378602793\ 3116877879601022855414344295415687688552513475969022334763018249628971861623666\ 6952575608925310961117473295018213416482587835259638606918019523765074857790833\ 3823732081084563589173585421432798522925337652855197138944000 -193 the error is, -0.3878899109 10 `The `, 66, `-th apprx. is`, 12429128166222960280402678514913120891718307981371\ 0095192420311368784279229711601951661638529667994440634593166181533900412877510\ 6090230449323757661809328772333675648050446034612552906363073985401169778625882\ 085698694213015595566714713679219126164985887768139/395631437195417734973660355\ 1264896858758812365946939847887558375844941673457532553354844282198953514250434\ 0792199090013962473796822553771242361736159361113864003630005168904767105725270\ 6128431385033735102018507224719448656693525884253407993087822315651072000 -196 the error is, 0.3758825100 10 `The `, 67, `-th apprx. is`, 96763603717626596745284658565116340569659420391083\ 3367666178962934298478781414469559846235756557832261926738562875511479605774363\ 5222407529231013113222609482051341188792276767123778673843699678973225392626499\ 8489751933885483054854683944391868419069479041438463407/30800811686091145908236\ 5412988375111647193562831520003020555876835816354859636812210225152727547842978\ 1431512023081326935581927879885691214123557986831769825430086032770982479330076\ 9539749712391168042069086659686889494150949092003862577534337299739438728871936\ 000 -199 the error is, -0.3642896593 10 `The `, 68, `-th apprx. is`, 13237548761370478662551953290692495661574225540490\ 2111264369464074283211967122871672127663478964045454419717394475162908734172849\ 7309236420355766849689933193083603552964891394164145186607613933777749808854167\ 07645883469667182341699574051275419264726998192520828759/4213642639584216243208\ 5677315881799660285216030483033870470097648914653750314627097904407510683124987\ 3444905734012612755499683814051276716281587114927171485411996899650454108696456\ 6241869105635669085309079513295455514441468128571532129566035734685544034284339\ 2000 -202 the error is, 0.3530946420 10 `The `, 69, `-th apprx. is`, 27763504250288451492601069797534771133978981754534\ 7049458655518566485798871708175055023608877375814698787089877155359754209881350\ 5223787495258754499869937675046153921032928325107170693596809694196578071940578\ 8854761620262424190632213901320894956292779293129059430925201/88373978779725055\ 5212855382856015172692119744062670901090136762343223916576021561704751751501435\ 3538761163590122620454597236943538171600626304151385775237105663536304859752538\ 9407210794222013613561851415607623612773330871965731945407249697544432998748089\ 71445403648000 -205 the error is, -0.3422815623 10 `The `, 70, `-th apprx. is`, 17544129184008991152580834864117279203219064860338\ 4648148891920468872470160663534157513835732476038284749791379412542134934648489\ 5142136329133907536668714406642883690749472108399852556704641042859815977301387\ 104671371634197774519302159136997468826563673348376945836781457/558446976375562\ 2281027372065527685769964932852011910271726264941536386799908549247682301320679\ 1062939521807755056443479158857566675868482802464933522241787111023299451359244\ 9247719789798578384942883266129529185457756095156277318401591515778273557837576\ 376303900164096000 -208 the error is, 0.3318352958 10 `The `, 71, `-th apprx. is`, 19265951228979097764701383845905984989220400699194\ 0313012415555282717897143192996595221723852868295166660217524115172337771757870\ 4530854605592465906940066029085579229623098235821386840721724995457638059230275\ 531179375817967598246467939470404145737908456610961782907916729741/613254274291\ 8798620052898281980079073039996617822531131063157947079769273815588290929925896\ 5570009322960001400141662690077871708931338593189391545396534968623293052306411\ 1497522999563332618223732758122760561768648339716480902997714328493792982554912\ 468480594679376642048000 -211 the error is, -0.3217414425 10 `The `, 72, `-th apprx. is`, 24871119263389950461939472713162734672917022022807\ 0816050345933233917246971035622985287517010153834866049354566615555692316444148\ 6950021495935326252425216410555315269450754163245471036879624325164099265296763\ 40367632974537127943476216964659187830835969043138790450617129819187617293/7916\ 7231419931388163999459219374107086719161330237236030250917778871836255516517174\ 2229446356213890289491673882939495344764586298194068051800650949559630565914643\ 7895289429109398697753301472389294192316879274924414620314380549030907832713408\ 1474814010876844984370121087966511104000 -214 the error is, 0.3119862847 10 `The `, 73, `-th apprx. is`, 22750005567581999540342420490805682495782743317368\ 2174738398201762224063050105134564980576627692270512739530848001942133920886518\ 2100725001092064116697430529171362848317414893399077157348360724087451173327641\ 9378943730178265765569979400212585494063544885721997263440225714805512241/72415\ 5168289763037009229403794785119067363403063835696069961556914633040671645607024\ 8322731297609195868747743550323293426714100002954873768324769669433657456465752\ 8839161562229428192515746310094655886451065216245993305169776897614381323094535\ 94876674118017579989838704156868608000 -217 the error is, -0.3025567466 10 `The `, 74, `-th apprx. is`, 23884418677970085659674476618968101658332862319843\ 1179027305829555106371906545981122068159630658054534364266695021436656250907996\ 1753063896791570324343633489630477566204790414805681024404255992979913434865520\ 597780727595253835874877162277459294395187973507096754244539460821569017371283/ 7602646590950662025854537171207193215861441385108642791421431102842454919895381\ 1048414451022932186899911006890545167248074469171102148444677272586892805734957\ 7208121638955957548441684728154478748516039202215680036924902446017481966391339\ 543609074701875715077168905625033100119506944000 -220 the error is, 0.2934403580 10 `The `, 75, `-th apprx. is`, 10680553137391157807001989561249651208027651040455\ 5579971421928020631980027504749581830624591894924589291595942172922971057649784\ 6424865152950005326300883255539092214431733081405394101177612220470381558097484\ 9581455533115618006198814067850943064172094837437504742155680558039788523812957\ 432793/339972565354290781247074402679405055781198542969217351408234472770333932\ 6732088878167240959930133736846674436389056003642529767073827699418607085267992\ 9433018894736532689475264625648787500804408153365745591352026559995233114178112\ 62921934865566577543307958907771288431231765989007088943104000 -223 the error is, -0.2846252193 10 `The `, 76, `-th apprx. is`, 60700915812372815858428091297324877706398045490033\ 5582318541900582793979674078073355412970710611527657415998717302679196487593712\ 8560593222142600671029725416804389251386441802348253159951937663945965153714133\ 4324481208822389752523622185636631973237535663613623240458031809071903880550382\ 311612707721/193217016034882504855087291764069490638195505363946860386663587790\ 9312083568840150635441855446244822822554698442315865417363313556788985003816267\ 0988607510582948068287307248704515012526178877170251092138230827806877273962830\ 060583764299387484692916950812642556043456796680670426515856675410280448000 -226 the error is, 0.2760999702 10 `The `, 77, `-th apprx. is`, 11072059497382144917432788350951598330718975890541\ 3361389440157562953661671480699853284069803013029785059479122031519244816524799\ 8029402165795087861498170520064079414815266837294629595261293261685567854915495\ 9375455508173434769127040935883952498919216044449486570286208037887972984106664\ 633129219650681361/352434599843186909743349148232874493356240938626526811485387\ 9173562312066352489344164318184798891175397138559373229619576560291643557596261\ 4360047576704700566167679738438477390740040873188688219238950848357867998717388\ 4127907295569065138332506418447703755384647127267181334266814298506309319908969\ 6768000 -229 the error is, -0.2678537593 10 `The `, 78, `-th apprx. is`, 89047702831265743427148246279829286670707206738232\ 1283776255833609508601212583817412028328434920472828127964407685815417591589693\ 9031348682536941831370726297846562538416672454952296197664666734701880607229171\ 0074663451815339536791134311079307188051595410828626874492090874423795813589731\ 98223141799958031909241/2834476415314821312825470125611787085287169123360159472\ 9799214486125308532363876699802178860336170211602010692072761609221660739160810\ 5813883747544238502156585437534357175760723670223125072416324522416534969054851\ 7645193531742099051940518947482820750764314906347599873941911720910004594903083\ 927083163844608000 -232 the error is, 0.2598762166 10 `The `, 79, `-th apprx. is`, 72437413862461142762357272979619255917223660317541\ 7021647878502107868697552531734392873472376677596898694603941377187820517215571\ 1705366949692345974574231948567524396932291361049511865177172552327042272960142\ 0206274099057869908221256885810264924092546673537781583686799825670148355161065\ 31727175873211825404279349/2305754496200814697035659268975570945631802967201453\ 0061744633427442888627488836806855266361823580693771405174757199855332831383349\ 3381462738919091242175631841491074407445448483208312303721048902897923088998905\ 9342927806707611368895025088004216147709465728310920115839521867412534849169114\ 314212318175047974912000 -235 the error is, -0.2521574276 10 `The `, 80, `-th apprx. is`, 36105000757293786950142204379290200382131039958852\ 8424041693243663874804726101388360367670447472442701029351071883454198399019878\ 7454751012404006074489172321862288380990077831372521746140282928964663293467373\ 9545740702551991117939237599578869447237756290354665337648709119585694673181762\ 9981025393342159357062926673743/11492578681719861327662682967835512642905185651\ 2602794467753951692624677163278933397207209394555019754244387034197381527515462\ 3774295099534014504410791253345503848874919714786417137950912853459216884504828\ 9350389587539952256741153081997222866793736809151034412407713278311353427832689\ 3488349375362896284290268528640000 -238 the error is, 0.2446879088 10 `The `, 81, `-th apprx. is`, 29682166869689061561696251300765726845639824721701\ 7637920000859904999498000078165267422141554124445885038963946044520794441717645\ 3745442721196186822119948821984948140985321388984692198871534413896971942979391\ 6848780246032918464605505398760042630737782647811285106251661576919486676196965\ 335004509451892522542560541/944812715797900742282674336990447008551538710281999\ 5568648425986155243155358153783311668914354090632848173217257647912891879963842\ 4687725353029925609489924789257137032380181769690641582017354987210850434680737\ 4626356389316947747944385413261794841881068776342532377291497297430623944277474\ 6215021137426378653696000 -241 the error is, -0.2374585855 10 `The `, 82, `-th apprx. is`, 38552943738424723416403098926035690908880502790712\ 0375948973758539970478062096901397919096162065275609147259018688443240403773738\ 8954420318581046585400591642821833034967210750567542990760259718656156438476519\ 5423480935383288660625934186444036677440340108239708873367659712371420297837473\ 20913042503800050285012927892848547746803/1227178313342805894969965055183618629\ 4645233585144899237611390104314164032860422170848600550849127327800411241348736\ 3742634794422351644595848768885724144266785313769856076723835611146065043751803\ 8845563894260159347498221518501735983308988843090908346952265292566781051862530\ 467347534328067105404632944428866940250521472073728000 -244 the error is, 0.2304607699 10 `The `, 83, `-th apprx. is`, 32694899040508239468715874803508189822725932886149\ 2994148934888735845093733180878717975545186790422042562545635329290176602161365\ 5827579828614835610735826419172281781602063143013773471384999475691519680941775\ 6638354611435438305699655874997345389848939860646689007276408460181286195877462\ 1495620892363464639711209419548633009808471/10407109592374704537862171182271467\ 0784458928975319729898184905689833105369582541267066703372785456429268422605204\ 2188622604165685657453520639559511400859820919608983324910398242260758187968440\ 6225534457440933559141770631838722514248061944344654066890387392675923480868392\ 6285088044144964651925873417235331183374492275488849920000 -247 the error is, -0.2236861417 10 `The `, 84, `-th apprx. is`, 17545978083044301480538460941393104421622379363585\ 2583686203286753284716765618384549243201153635128335224025253563085877057677062\ 8617480349182225933511527590913712982837982990937465067165779964305676346806221\ 6540657896636371884326140140175337382310624943849356891014111537165724584348892\ 0907332361066004669029720652033340993870132728880343/55850582865971172318894918\ 5074596314669766948033514083974687583757774725867700556046601623524977895472837\ 2740187416231964794704252002557504408720678034582649475612483456922981027624302\ 1529450539262818716442338838720304512046449282122159709901311432003313768862791\ 777663184634417138471931027534366514718729010676790407595542404724162560000 -250 the error is, 0.2171267294 10 `The `, 85, `-th apprx. is`, 56663943201341843252161398953328426376899337198274\ 5718174294431827128579306188911282052903555063298736680875103673765177858970761\ 5994729297658559047303868279608329456387857662597412471771984738243824277769522\ 8917779106666693715140442643239308084872430012927471119934974563765392674249503\ 2958370060205960871300807084185911596852498587/18036693311143901345577898217132\ 4446645252072759901722515029102340704078792685498437206155255437637255804914522\ 3838754466099722518241296578235679031671043490218604127407797138558991120714402\ 2470308452972380078240098164985178205333501055938034664580462596651775928931862\ 9315368146906703547526042657321132541108896029188885934243840000 -253 the error is, -0.2107748923 10 `The `, 86, `-th apprx. is`, 24302720883731172161987818902238541594239390728368\ 8801696342548250007741249902154519600639120573988053958241054563146921188323478\ 4167228043091255084639465365999078341779018930874313064956105092675426350936920\ 4493912118231015598698023302292755276135877544561769943653107820345874915829374\ 732811246559898297008609919963843462857113666230492455653669/773579631845690205\ 6944595386247328483579078153551322701820786400618752782586854889413863820687048\ 1928821795854210695439130564529558450380482572654011022555272636532273883851848\ 3791312500053001438143727682686915866712149407326704158441328087222340645207767\ 4095642351969326065280602398506784755655624164009257294365256378922942298779611\ 603927040000 -256 the error is, 0.2046233046 10 `The `, 87, `-th apprx. is`, 12888735771703529593227223908241583964643012670323\ 3646432893551620728568756985030505794336700199583678886111457286663312630838168\ 8539982725807607425405718713341903565970470108931285338638055502036196915549106\ 3919976687309689524331832057666616340102344498921132626335494166323674054244848\ 03717730908982543258942681970855226253248540207317253199407/4102612016543901933\ 3720220901556278714792175943437150330734146924118555348113692506655813872184290\ 9945919839798314283378702298976498983653916194608762887022954620115254592311312\ 9863465757162712911070124682805944302243258494422012969021447245974753742104322\ 2481839704896142427974154118726047195813109477177642831474706734513876538932909\ 3099520000 -259 the error is, -0.1986649399 10 `The `, 88, `-th apprx. is`, 24457369556584693319899539349622307288121144900221\ 7697931885556711904372106062656695118982735784112233554482554379670028963748289\ 4312415862722296764892862472988579773466000285696914077846825970394499617733020\ 4130904897043742263961784642552350728781301131020960999436591349558021546439064\ 590394555730988155547744476686756345274218231258713940402662127/778502251991138\ 0654854236292558055136532224528399829206300900868408017020324305032315140980640\ 4253071773402852071274140513599939054999576026956192237679222823667097895662142\ 8003424026245538778837046704521414567610405595360535344855470348748714803138861\ 0466097070849473401680788701046247692593133207257576674368702745852046259770477\ 103293787013120000 -262 the error is, 0.1928930560 10 `The `, 89, `-th apprx. is`, 42375856973434055495893076503865069799798512596985\ 5223663865931756449665236374808058103144527810533100756873955003058731699377161\ 9055511532571863889761146679562202924919854843452296970475553306168866409198057\ 4436414675331830744589140587930524457830145470277245662283205321212456162936092\ 5743348291897648873949044413678694109446450567884537647571868217/13488654210154\ 3864435512326822807056026733158544510610162344836866512249008875417978106722788\ 4348024770275583169715661983180716111180353489760538631356656103724315285628601\ 1029717399758990426112160385213488636866240659950379575891185974883217020387614\ 6753677066734153882312167370745202794032947411191385189504789572101271761006132\ 255867119100493824000 -265 the error is, -0.1873011821 10 `The `, 90, `-th apprx. is`, 67876495076736992687012041652210163675418702958308\ 7717202696792334137298344730738152690436499115755474292020114152323393238560114\ 8815080062959925623921603248133052553217065668190147384830144350771362421087118\ 2260624356467357460886632569123122366287074034333300560763629251733606107194363\ 52007371900801065360777984878116986111128949081801856065800691748327169/2160575\ 9422430780005969099916948276199169724621468468466944754391321655132924360310596\ 2344059655021181075367762538833109865292855141046611428350929305631884589958692\ 3144124148790132262067742000686825427248829164142152736256735433803745430942019\ 9753926643884822547855251970203080350718156596597746707677639087161740631545595\ 21877232806505917895314738511872000 -268 the error is, 0.1818831051 10 `The `, 91, `-th apprx. is`, 10274357886352399524623506936408233196342325774110\ 3172351313472355419940528391873838270405019550363828634939465700109583102847310\ 0210114224266986636547291102191087323318541097984992835724815534358865174897186\ 9483239771010532634605787119410426311865348680144345916460852452457898095901204\ 9557991924464480126574321978328974513691977844376594304593058725728938009/32704\ 2968941636438406142796637596012362169094796543764794384808576005685064686632911\ 9724744818778110088277672026640442231434643428345526602462827750963143684635374\ 7320855268694318002030035926284080578572461224610909638260391426882208272944153\ 9444696279093751313934903708770021363624554833535869050164636263235142979243816\ 48341678534481636946668027305053388800 -271 the error is, -0.1766328580 10 `The `, 92, `-th apprx. is`, 11353576438734855570689960105008554011286123673422\ 8649575095439491633251081494156266242328362803934045194753507177249093711970391\ 4656184622383990912850218559565239135759920654917336483189350158087920372868387\ 4653759276557478982544778998433297491663684905506708011525900394127972038428903\ 7629560249772267553179946323069661966985098835204457485880674742273775929097743 /361395562399265929896324035996409097500691336513972721848386988868829322223881\ 3168330460632014542562771950358696318754283424538374058940716785523177924312297\ 2694744939437906117969164963270899980960402545712551644039586663262942361915429\ 9342078747967176249758951950625994339224407659678073250476735193929656325362397\ 78358696683488447743588091546036893176196759552000 -274 the error is, 0.1715447079 10 `The `, 93, `-th apprx. is`, 92548286837711356741183668762870158762515689326538\ 4346368301678621583217458003701212678951982900999268231060980875788151217481405\ 6736624942996812822963630718690544565521622186811947055253050249855128163561733\ 9278923785641856909697126530591147493038194276723947366021945135133860710000481\ 2573555159799173808909168911576646365033200850247428990122585286051845519932516\ 029/294590346498167142527644067859590129589421299196135505688085590573016329495\ 8926351360546258460337078517476810924550107317049776508930097046181799268405079\ 2921802560321020949041351813202000055943962696409611566774184924104466845180307\ 2181749042587685623143521182927546052989086571681691203209873522842520745803080\ 105579505455370742858042280963130914698970830775255040000 -277 the error is, -0.1666131456 10 `The `, 94, `-th apprx. is`, 27053394517892681275998642156160937751279881599622\ 0038506654072125717418164532488055097583782104455423587144171180305340772975666\ 5447279991538401281262977716399032754264886362270022727678831578710469391222663\ 3415818869672826990777355842750961968736511424692869647464438536080010830643967\ 1041689894229241179280906308224538465876475205865258744314324754915553774808497\ 9453/86113629298756058830171167785319748729316942533668483327898998388882799743\ 8343014164223445936594173995965280483542572488820236702496257225405083093581873\ 3944171508530524169387964779757021364748158136736326020364717940931158294451574\ 2846438835420074820029440486780608768799330225988384158144018440560937687857635\ 3049488216288332399389510295311053110911496865790191206400 -280 the error is, 0.1618328751 10 `The `, 95, `-th apprx. is`, 25387231408994432716206501582979405725100943815658\ 5638586108186464342340100486169407384636547011395328182030315322861254460716679\ 8423546183783533800072617507950334944946858254234854681427684083382427937879511\ 5128337087918360405710709589975407860554065595894456426259881068741841927475903\ 7024131953620205580095624838999035687252253022002739332502195380313532719358922\ 3448267148809/80810067403185735616741527412315865310563207142752035460124521271\ 0582053173510216525027960559423508837610470587386176711507400405474752860219696\ 1062685076548954570871721673369907274299584920787116810818916010238796710620987\ 1639708054501799502746165023017208307552788974976955515996722854031452540044592\ 6855847436546693988497977397992529104718950737469335142394748199889797120000 -283 the error is, -0.1571988041 10 `The `, 96, `-th apprx. is`, 83022339642949953645847149736722571778454110503719\ 2618729834435703150894290213890642805533821498545458034002818381433788787614115\ 0876617672056263574349476679199503350363013737328991173392069596840551139571093\ 7297293078394138933171390929521177801898327474317887183412113466242321144909215\ 1539639185712814049836726902267087303864133590782011841418176247312581093061158\ 048138937331563491/264268314824594120043292812604851835373216222526513416443117\ 6144414653853570140050488807437739849135541107265333693770529039961101999755793\ 6048589945635244733453195847539085121205564705474591354860879412451893323322554\ 9411770631038868153964805860618704871795295691432617528639006633122946167817354\ 5554287841066520892460548945625606011088379416101459701718556065001353207610133\ 70880000 -286 the error is, 0.1527060345 10 `The `, 97, `-th apprx. is`, 49359375365847589628507563241910841502659045385414\ 3424104221879351638930120978727171230477529812807104346778238115836807240136204\ 3607114100339700453187462306930329726270510511021376783555753877496608919698126\ 8190031275514015412612053513566887740034865006215556364512983084226879993184306\ 5818467656989850326343156921162996091279920762185113279230093688859389577370643\ 3168874911938195217/15711577154805947293198892999397831774923245729896617961969\ 7269210902467388037232689217379696876968136467392884292262450984328937392329231\ 1666638824111595409543584534373222170096674589130169064144463221318428970238161\ 2742371675798482708216188848432096438080954689154704839007365941234887658883516\ 1575532269300907999934568574032897357377988807472907096328736028552033577421196\ 2306560000 -289 the error is, -0.1483498539 10 `The `, 98, `-th apprx. is`, 59858736662691119561775097517460361647698932405296\ 4300169639918901349078002446555867016665337621019911060345766594584626656908187\ 4596928797507293520092677361632877030751882671438342494350312420228343655871273\ 3133419023860448291184074620891736282041995289053282519738381681508411869631736\ 0504227804419963173019521371672392693197491524026922894382634011078916824120692\ 94358080527915060267427011/1905362765420670812955938754759893991851049671273422\ 1993476037785702387694948842566617104173284005074277052849967785973811653546166\ 0114219223695697362862117211137662062795135545244400435343988795677027376809472\ 1001974659921294460020500627026302784877464160538865780277396018908260217123737\ 1643127544666915817355775417387824848710387491265385395535279345069807657696679\ 114083585557297889280000 -292 the error is, 0.1441257278 10 `The `, 99, `-th apprx. is`, 46570097123573691019061025868584161361909769411320\ 6225531979856905249582685903420464538965632669153490804949006410586839539074569\ 8436410604460674358632102987350378329924964718379030460604543062937651364267850\ 6377800000563428770541210055053770827428672334883453800356460948213544434573490\ 6472287093283208799126536396899468085984246741155407347693840347348783949138319\ 71161950488965424556316230343/1482372231497281892479720351203197525660116644250\ 7224710924357397276457626670199516828107046814955947787547117274937487625466458\ 9171568862556035252548306727190265101084854615454200143538697623283036727299157\ 7692939536285418767089895949487826463566634667116899237577055814102710626448922\ 2675138353229750860505902793274727727732296681468204469837726447330464310357688\ 016350757029563577757859840000 -295 the error is, -0.1400292910 10 `The `, 100, `-th apprx. is`, 8032109540696604595806455399416391833108945372339\ 1625224785269046819371841022128586605559302799407474988678244647829212342651821\ 5527704142287504455300225328034154526682790254448175776725829888499128263209262\ 1692454934673108332595813311818479299736904462107081664058587260077346706061494\ 2990030436331012673550712782014142840678419628264571138773152877640033467449836\ 1710325251397814967030758037/25566998737148753758277673445828300676926032772171\ 2426614452751737138317746107158112022227518643668315165252855806823120357375326\ 7571488607878276716603835877263020989203267735037370007847437171362705980211831\ 5738751129537317147770828516690171979392841263713845137566943499823077935309860\ 9233939351921662106213030536138180929572500404813741292697750015799856449231125\ 093843758197718028124160000 -298 the error is, 0.1360563408 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - Pi | <= -------------- | bn | 0.7480429137 bn since , 0.7480429137, is smaller than 1, this sequence does not provide an \ Apery-style proof for the irrationality of, Pi but this sequence of rational approximations is still neat! since it is effe\ ctive! This ends this article, that took, 1.083, seconds. to generate. ---------------------------------- if the input is: DelSeqIntV(x^n*(1-x)^n/(x^2+1)^(n+1),x,n,0,1,Pi,A,100); the output is: Better and Better Rational Approximations to, Pi By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / n n | x (1 - x) A(n) = | --------------- dx | 2 (n + 1) / (x + 1) 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: Pi Pi A(0) = ----, A(1) = 1/2 - ---- 4 8 Note that these are linear combinations of, Pi, and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients (n + 1) A(n) + (-4 n - 6) A(n + 1) + (-4 n - 8) A(n + 2) = 0 and in Maple input form : (n+1)*A(n)+(-4*n-6)*A(n+1)+(-4*n-8)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , Pi, and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , Pi, yields the following sequence of terrific approximations to, Pi `The `, 1, `-th apprx. is`, 0 the error is, -3.141592654 `The `, 2, `-th apprx. is`, 4 the error is, 0.8584073464 `The `, 3, `-th apprx. is`, 3 the error is, -0.1415926536 `The `, 4, `-th apprx. is`, 19/6 the error is, 0.02507401308 `The `, 5, `-th apprx. is`, 160/51 the error is, -0.004337751629 `The `, 6, `-th apprx. is`, 1744/555 the error is, 0.0007496887525 `The `, 7, `-th apprx. is`, 644/205 the error is, -0.0001292389556 `The `, 8, `-th apprx. is`, 2529/805 the error is, 0.00002225324250 `The `, 9, `-th apprx. is`, 183296/58345 -5 the error is, -0.3828497669 10 `The `, 10, `-th apprx. is`, 3763456/1197945 -6 the error is, 0.6582901345 10 `The `, 11, `-th apprx. is`, 4317632/1374345 -6 the error is, -0.1131432533 10 `The `, 12, `-th apprx. is`, 54743776/17425485 -7 the error is, 0.1944054135 10 `The `, 13, `-th apprx. is`, 1013549056/322622685 -8 the error is, -0.3339548126 10 `The `, 14, `-th apprx. is`, 30594128896/9738413685 -9 the error is, 0.5735732051 10 `The `, 15, `-th apprx. is`, 35618973952/11337871545 -10 the error is, -0.9849810490 10 `The `, 16, `-th apprx. is`, 10392576224/3308059755 -10 the error is, 0.1691284717 10 `The `, 17, `-th apprx. is`, 3111643512832/990466892415 -11 the error is, -0.2903785159 10 `The `, 18, `-th apprx. is`, 123968232030208/39460313827935 -12 the error is, 0.4985149181 10 `The `, 19, `-th apprx. is`, 48501417558016/15438480702645 -13 the error is, -0.8557822278 10 `The `, 20, `-th apprx. is`, 1083228572868608/344802363740835 -13 the error is, 0.1469008045 10 `The `, 21, `-th apprx. is`, 4080033616887808/1298715036217599 -14 the error is, -0.2521531639 10 `The `, 22, `-th apprx. is`, 188557135970304/60019600489849 -15 the error is, 0.4327996303 10 `The `, 23, `-th apprx. is`, 3781715948011520/1203757572990973 -16 the error is, -0.7428376676 10 `The `, 24, `-th apprx. is`, 51359350521741312/16348189019048887 -16 the error is, 0.1274933392 10 `The `, 25, `-th apprx. is`, 11656085634585460736/3710247291693415473 -17 the error is, -0.2188110551 10 `The `, 26, `-th apprx. is`, 689480139696779558912/219468344792865993825 -18 the error is, 0.3755265070 10 `The `, 27, `-th apprx. is`, 816318089349070585856/259842118110472122225 -19 the error is, -0.6444699714 10 `The `, 28, `-th apprx. is`, 8704683425259812552704/2770786790360380066275 -19 the error is, 0.1106003583 10 `The `, 29, `-th apprx. is`, 165125781784343361880064/52561168805783790427275 -20 the error is, -0.1898029711 10 `The `, 30, `-th apprx. is`, 668354038174457037586432/212743697821788305247075 -21 the error is, 0.3257188197 10 `The `, 31, `-th apprx. is`, 586398465775082011623424/186656428899215824574025 -22 the error is, -0.5589548425 10 `The `, 32, `-th apprx. is`, 62071707951057912930762752/ 19758038293134738732210075 -23 the error is, 0.9591912020 10 `The `, 33, `-th apprx. is`, 75529446903693013937121394688/ 24041769647439184012627430175 -23 the error is, -0.1645995872 10 `The `, 34, `-th apprx. is`, 179587785206369053071833563136/ 57164567469038379400282206975 -24 the error is, 0.2824540637 10 `The `, 35, `-th apprx. is`, 213600784656597000231439040512/ 67991241452809771944641368575 -25 the error is, -0.4846886297 10 `The `, 36, `-th apprx. is`, 254163521283049527402720919552/ 80902761531678953056929313575 -26 the error is, 0.8317141676 10 `The `, 37, `-th apprx. is`, 4840805276130436433254081888256/ 1540876176482972594682934491975 -26 the error is, -0.1427190471 10 `The `, 38, `-th apprx. is`, 426577879287040123519531602673664/ 135783956204380537892249860223775 -27 the error is, 0.2448987788 10 `The `, 39, `-th apprx. is`, 1299650133582937701611579899904/ 413691486099533242160441406525 -28 the error is, -0.4202312950 10 `The `, 40, `-th apprx. is`, 302779142809200434162121387802624/ 96377594486422291933827928016025 -29 the error is, 0.7210866822 10 `The `, 41, `-th apprx. is`, 23099314802942710841421068087853056/ 7352740265848245332158839252232725 -29 the error is, -0.1237325797 10 `The `, 42, `-th apprx. is`, 2258601801834136133882621891503456256/ 718935282476328405580165727291362725 -30 the error is, 0.2123138619 10 `The `, 43, `-th apprx. is`, 10401447772383710058682038630744064/ 3310883656574101768087551211076775 -31 the error is, -0.3643094908 10 `The `, 44, `-th apprx. is`, 19744179324824588268101438584100225024/ 6284767473677267638687922870355272775 -32 the error is, 0.6251160094 10 `The `, 45, `-th apprx. is`, 377007404451960464990387692952593891328/ 120005184001549872126533838222292079175 -32 the error is, -0.1072627415 10 `The `, 46, `-th apprx. is`, 900080224856949698989249134541869678592/ 286504433930496373957325787825534468075 -33 the error is, 0.1840498308 10 `The `, 47, `-th apprx. is`, 2039283756751106417221951931073691648/ 649124180507897745149135166490528425 -34 the error is, -0.3158059607 10 `The `, 48, `-th apprx. is`, 367832870042189999058590903013157634048/ 117084838997786563989398648673337138425 -35 the error is, 0.5418807167 10 `The `, 49, `-th apprx. is`, 4611896713650047882843235774928637497704448/ 1468012318013344979546937184261656155195925 -36 the error is, -0.9297916423 10 `The `, 50, `-th apprx. is`, 540012552519162474158649153000704297924558848/ 171891334130192889631238384068788965571815925 -36 the error is, 0.1595387681 10 `The `, 51, `-th apprx. is`, 645344067088708499084128752426733841265197056/ 205419396544391373693847962022223440202351925 -37 the error is, -0.2737446024 10 `The `, 52, `-th apprx. is`, 771373182244496277073168949216840445253910528/ 245535709845474029253525527117296976467970025 -38 the error is, 0.4697033994 10 `The `, 53, `-th apprx. is`, 14755032535683874348489489993264042894371586048/ 4696672727071662314200961896099736354055521025 -39 the error is, -0.8059363019 10 `The `, 54, `-th apprx. is`, 81311724350457272080601510094214353440636665856/ 25882325723401814746111635581359472147239358175 -39 the error is, 0.1382855089 10 `The `, 55, `-th apprx. is`, 18183908462753037461251678239456505593774211072/ 5788117833155387347995425860574361666132039975 -40 the error is, -0.2372748043 10 `The `, 56, `-th apprx. is`, 26228720283579673725940711936265640283633352704/ 8348860968212727730116755811831078347706586575 -41 the error is, 0.4071229899 10 `The `, 57, `-th apprx. is`, 34139870611995353822646146874986781853546484596736 /10867058328833581053807279740470880703659580801675 -42 the error is, -0.6985519876 10 `The `, 58, `-th apprx. is`, 4299941933402074018786770494275071851386296598528/ 1368714027418122988596952409718219289287226301025 -42 the error is, 0.1198590930 10 `The `, 59, `-th apprx. is`, 109485130993621528545197550892625386733320536064/ 34850199585397081227085723833273635432687044575 -43 the error is, -0.2056564969 10 `The `, 60, `-th apprx. is`, 363378525701292466007428838346262042668366114586624/ 115666977157612058168528007620500040316837810423825 -44 the error is, 0.3528686809 10 `The `, 61, `-th apprx. is`, 6959774862892069622030813282353341853489967214362624/ 2215365144471982007495298655524627284509535212176825 -45 the error is, -0.6054566663 10 `The `, 62, `-th apprx. is`, 44198019597676556867360619546563484554482685562585088/ 14068666587685374383818460042448692207727648246818775 -45 the error is, 0.1038848754 10 `The `, 63, `-th apprx. is`, 1217205804085873955172095555813426474527965205332230144/ 387448640960823945013809917397536196623560463022260325 -46 the error is, -0.1782464633 10 `The `, 64, `-th apprx. is`, 595445305153146168699229906543009109487958441852928/ 189536127311970461998096281009952867653720792735775 -47 the error is, 0.3058362013 10 `The `, 65, `-th apprx. is`, 49658565871021680189297064673937398677999829268060700672/ 15806812450455182893621145284198377096825185369685045475 -48 the error is, -0.5247546076 10 `The `, 66, `-th apprx. is`, 118965238722944010969584526187920220914487302872601985024/ 37867811597727794830651905066441720985619896497880775075 -49 the error is, 0.9003742338 10 `The `, 67, `-th apprx. is`, 427551205780047151120906412145988178496996035880336490496/ 136093775649589272975017591072199090009314624778452488425 -49 the error is, -0.1544860644 10 `The `, 68, `-th apprx. is`, 34320934501158864092534710454010264805317357878236174876672/ 10924692754785212496955939107090104674407607424412188846475 -50 the error is, 0.2650666192 10 `The `, 69, `-th apprx. is`, 433748097175521538887359754418787561778614984380573024256/ 138066307444376037091898000125535304864161749397873675175 -51 the error is, -0.4547998042 10 `The `, 70, `-th apprx. is`, 525681264041168540926376680618762242661741141423142947782656/ 167329543325895570102010146581385960470973964514448882948925 -52 the error is, 0.7803420704 10 `The `, 71, `-th apprx. is`, 630025718287668208586323055695956203914283386270281736650752/ 200543414681008632531091596408519291624863204755354322360225 -52 the error is, -0.1338903623 10 `The `, 72, `-th apprx. is`, 26808117630000529085773683668696625811268724353479932010037248/ 8533288871607140535113567555987432183897450157127667589001025 -53 the error is, 0.2297275992 10 `The `, 73, `-th apprx. is`, 64049145087364837635107963952449083695424266022072703613140992/ 20387476082928196895066413077924418540727601747573588235039725 -54 the error is, -0.3941636864 10 `The `, 74, `-th apprx. is`, 3239872552061714492221802333981472085079222135848716515068805644288/ 1031283463296751757023872976871378152714487677690270530114864930525 -55 the error is, 0.6763004014 10 `The `, 75, `-th apprx. is`, 3884473895732202411260267914372487390360664547360142426813136633856/ 1236466443634423301924730433240239897835486447618071941010090266925 -55 the error is, -0.1160385499 10 `The `, 76, `-th apprx. is`, 113603560144777495201623838308750420863686217330579468724016775168/ 36161136299757542601084249544233615679337158499127272979634707675 -56 the error is, 0.1990969326 10 `The `, 77, `-th apprx. is`, 73824992343154353323088106935516707191791851479967202771815693811712/ 23499224910268680005694780884770374855618007830763012591721466106575 -57 the error is, -0.3416067854 10 `The `, 78, `-th apprx. is`, 131376803290344123984617741973998061140521556700763134262552551227392/ 41818535302539693907531072146477337653270040473696560480166814032075 -58 the error is, 0.5861220572 10 `The `, 79, `-th apprx. is`, 4884676931088653047956905689537569052672240057167876770772119246077952/ 1554840957979417060998540119299691931367064335540982489789360627556825 -58 the error is, -0.1005655683 10 `The `, 80, `-th apprx. is`, 3127462274861299237927827848999871612123081929143848926806245797527552/ 995502160755199217942486030253860589349434390583220920667065191524325 -59 the error is, 0.1725481239 10 `The `, 81, `-th apprx. is`, 2090307685070546717719227509338046188361037389529539115501152334465990656/ 665365601323908680834553546028436142597689749135529189753531245299799725 -60 the error is, -0.2960539538 10 `The `, 82, `-th apprx. is`, 767345004526392824629291234276367085993928068495152226723889371911183925248/ 244253501054496437682024684735823602486954505032012681641972519794476455925 -61 the error is, 0.5079619656 10 `The `, 83, `-th apprx. is`, 19587755433170902424399177180886642634842840105494212664037331149787234304/ 6234976202528557294207660314980495111648514969026294804911395102426061275 -62 the error is, -0.8715478603 10 `The `, 84, `-th apprx. is`, 91682094284911227055508279821553482079478637913204430252644483657479271481344/ 29183316996921657880771708908421253646399684145555718079704371165443819940275 -62 the error is, 0.1495378091 10 `The `, 85, `-th apprx. is`, 76530024241363818087409946678173670448826468656081806135369061559564465340416/ 24360263305911257968050082101817864113110529248099069667577404160884168235725 -63 the error is, -0.2565727461 10 `The `, 86, `-th apprx. is`, 42245008421840177116125179620958234940235073922807\ 45383270078062351213496631296/1344700382258922515502226559014413179506367234507\ 935604087467145841588340377975 -64 the error is, 0.4402200075 10 `The `, 87, `-th apprx. is`, 50698014035193796107362035888035775767760204087363\ 22043801336418668639753863168/1613767907728675927200108442444875453701859777795\ 875985459713413217675320380675 -65 the error is, -0.7553161758 10 `The `, 88, `-th apprx. is`, 30423252270769121674043146357274118707622481281018\ 84382351277462962002464866304/ 968402196764927021759237076380986264968952680007263725460607780913350945580775 -65 the error is, 0.1295947944 10 `The `, 89, `-th apprx. is`, 57000008449224490982640132089470056734214799672536\ 12162874967832588854509961216/1814366620194775434363360061994222948892081323000\ 722435825766105519475420797475 -66 the error is, -0.2223545903 10 `The `, 90, `-th apprx. is`, 29371819982971057511477937571492242352991865206526\ 09993334402288046959771756003328/9349340675790305776734656221753490154272002887\ 00982198099097419929813493616631675 -67 the error is, 0.3815086774 10 `The `, 91, `-th apprx. is`, 59938793306755290513022531093088488097341432433284\ 125274133464393368264249794625536/190791104754670306261090836965160716280561761\ 16978480265537583750227960101161216475 -68 the error is, -0.6545796916 10 `The `, 92, `-th apprx. is`, 71955306688021942211456815121253697173750972300679\ 523733620740689865824475858075648/229040854821840161277249744507337964916482203\ 18842617085758858159547077477620348425 -68 the error is, 0.1123105171 10 `The `, 93, `-th apprx. is`, 13821770625535114037425323766891263296717794005859\ 80916802232015316843643691102896128/4399606234672543343357942289283274797433200\ 28840932154676159226420216292435422531425 -69 the error is, -0.1926984076 10 `The `, 94, `-th apprx. is`, 33189447558578898834408678023304418506361280077602\ 86011573739655713519579538554617856/1056452927487413846474489392983662988928680\ 596669253922475354585948028635742814459725 -70 the error is, 0.3306249556 10 `The `, 95, `-th apprx. is`, 39850273200154360039973286705738370996193160729024\ 98454398387728951567128920118525952/1268473592673409799914672232863188481626553\ 373719984967488160189517880160002130847825 -71 the error is, -0.5672740908 10 `The `, 96, `-th apprx. is`, 14588584194612295078260845722849582458407393359167\ 292208364265798947073292750553088/464369057456968714647262601863566725781557406\ 4129205603523811639256044546218148425 -72 the error is, 0.9733075312 10 `The `, 97, `-th apprx. is`, 73548961354986951483517238313155860272414926175198\ 7465902351960598423790845294853226496/23411361517841915056544267058432582349659\ 4246147510505736831573718807990887717185200725 -72 the error is, -0.1669963746 10 `The `, 98, `-th apprx. is`, 10079345712870690646961295917184730593093062097183\ 235384164721440590558825825775839608832/320835538667094798980533873526835746671\ 4404968168985580824058335595951729664335790032325 -73 the error is, 0.2865258643 10 `The `, 99, `-th apprx. is`, 20578190210233194775388180060383784086293378191830\ 0384516231026391980979365692990726078464/65502413836877236601402795892563911609\ 376059854500579331171437038881859031121449691381525 -74 the error is, -0.4916097119 10 `The `, 100, `-th apprx. is`, 1074552694180515569091644141808760106149417561810\ 6897414296601923415061141184538965180416/34204074578308553548494066660767747819\ 10448645107900552715681530568984194102177712406475 -75 the error is, 0.8434841302 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - Pi | <= -------------- | bn | 0.8286773817 bn since , 0.8286773817, is smaller than 1, this sequence does not provide an \ Apery-style proof for the irrationality of, Pi but this sequence of rational approximations is still neat! since it is effe\ ctive! This ends this article, that took, 0.358, seconds. to generate. ---------------------------------- if the input is: DelSeqIntV(x^n*(1-x)^n/(1+x)^(n+1),x,n,0,1,log(2),A,100) : the output is: Better and Better Rational Approximations to, ln(2) By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / n n | x (1 - x) A(n) = | -------------- dx | (n + 1) / (1 + x) 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: A(0) = ln(2), A(1) = -2 + 3 ln(2) Note that these are linear combinations of, ln(2), and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients (n + 1) A(n) + (-6 n - 9) A(n + 1) + (n + 2) A(n + 2) = 0 and in Maple input form : (n+1)*A(n)+(-6*n-9)*A(n+1)+(n+2)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , ln(2), and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , ln(2), yields the following sequence of terrific approximations to, ln(2) `The `, 1, `-th apprx. is`, 0 the error is, -0.6931471806 `The `, 2, `-th apprx. is`, 2/3 the error is, -0.02648051389 `The `, 3, `-th apprx. is`, 9/13 the error is, -0.0008394882523 `The `, 4, `-th apprx. is`, 131/189 the error is, -0.00002548743825 `The `, 5, `-th apprx. is`, 445/642 -6 the error is, -0.7631144624 10 `The `, 6, `-th apprx. is`, 34997/50490 -7 the error is, -0.2270690510 10 `The `, 7, `-th apprx. is`, 62307/89890 -9 the error is, -0.6734173308 10 `The `, 8, `-th apprx. is`, 2359979/3404730 -10 the error is, -0.1993185754 10 `The `, 9, `-th apprx. is`, 25786503/37202060 -12 the error is, -0.5891877401 10 `The `, 10, `-th apprx. is`, 1277351989/1842829380 -13 the error is, -0.1740116997 10 `The `, 11, `-th apprx. is`, 336763603/485847180 -15 the error is, -0.5136071587 10 `The `, 12, `-th apprx. is`, 432764726879/624347525340 -16 the error is, -0.1515244169 10 `The `, 13, `-th apprx. is`, 268565276291/387457792260 -18 the error is, -0.4468702100 10 `The `, 14, `-th apprx. is`, 58696128257959/84680613157140 -19 the error is, -0.1317532654 10 `The `, 15, `-th apprx. is`, 109957709376551/158635442025060 -21 the error is, -0.3883713332 10 `The `, 16, `-th apprx. is`, 428880251564129/618743412066780 -22 the error is, -0.1144608536 10 `The `, 17, `-th apprx. is`, 1907848232481809/2752443183770280 -24 the error is, -0.3372909193 10 `The `, 18, `-th apprx. is`, 864932284409627477/1247833517422532040 -26 the error is, -0.9938042024 10 `The `, 19, `-th apprx. is`, 3811914076838871501/5499429534950270920 -27 the error is, -0.2927884039 10 `The `, 20, `-th apprx. is`, 411013410547379951933/592967009135564623560 -29 the error is, -0.8625224691 10 `The `, 21, `-th apprx. is`, 467124680523046217643/673918460067440131544 -30 the error is, -0.2540714155 10 `The `, 22, `-th apprx. is`, 7973183151735023566051/11502871793107553955096 -32 the error is, -0.7483665878 10 `The `, 23, `-th apprx. is`, 890473949822518830935/1284682351449755554296 -33 the error is, -0.2204193612 10 `The `, 24, `-th apprx. is`, 458115624245374740964181/660921139252553908024296 -35 the error is, -0.6491795396 10 `The `, 25, `-th apprx. is`, 1618456036804583660014977/ 2334938498194778488708696 -36 the error is, -0.1911886576 10 `The `, 26, `-th apprx. is`, 1617747236950397613095136509/ 2333915916160161458446721400 -38 the error is, -0.5630458124 10 `The `, 27, `-th apprx. is`, 9247483926630414665038090827/ 13341299201650097971317224200 -39 the error is, -0.1658102702 10 `The `, 28, `-th apprx. is`, 129844410561229599436416626569/ 187325887203836489047256717400 -41 the error is, -0.4882775139 10 `The `, 29, `-th apprx. is`, 649038289689489988223622997/ 936364321882081635649306200 -42 the error is, -0.1437841224 10 `The `, 30, `-th apprx. is`, 1358092863869505703636243069755487/ 1959313839771225945664351975480200 -44 the error is, -0.4233944625 10 `The `, 31, `-th apprx. is`, 864838797362902764880166178139009/ 1247698644123834943015687425113400 -45 the error is, -0.1246724325 10 `The `, 32, `-th apprx. is`, 65887938730505508697549828728680153/ 95056202460896160608424125233207800 -47 the error is, -0.3671026885 10 `The `, 33, `-th apprx. is`, 1764091930992948736160602173259772489/ 2545046680515762590504810225769416400 -48 the error is, -0.1080929376 10 `The `, 34, `-th apprx. is`, 91134180279964412977949995280592563411/ 131478829945385421408430906724602863600 -50 the error is, -0.3182734103 10 `The `, 35, `-th apprx. is`, 7584840124794518855041879883180286793/ 10942611233976678692973010918410766800 -52 the error is, -0.9371244372 10 `The `, 36, `-th apprx. is`, 3006734553512037793294653443769253974527/ 4337801029621308498395624689114985281200 -53 the error is, -0.2759234136 10 `The `, 37, `-th apprx. is`, 49233702991190668481864121842748552071/ 71029219149990901481065071020380647600 -55 the error is, -0.8124089377 10 `The `, 38, `-th apprx. is`, 408478119033711348330422850883838367340303/ 589309356641587055556194272828611170026800 -56 the error is, -0.2391971995 10 `The `, 39, `-th apprx. is`, 328089064840188419705531916740375295673/ 473332466814837398772474316949221918800 -58 the error is, -0.7042601542 10 `The `, 40, `-th apprx. is`, 2385519501057805841378088138240236618849249/ 3441577154120010784623089051860014165704400 -59 the error is, -0.2073509862 10 `The `, 41, `-th apprx. is`, 77803107599528256582595778037032562672857041/ 112246157499589837746425768547118431907011600 -61 the error is, -0.6104854999 10 `The `, 42, `-th apprx. is`, 4238177859948983928967324788845644406745791813/ 6114398180953798798738514208891117479040958800 -62 the error is, -0.1797385077 10 `The `, 43, `-th apprx. is`, 5036515744909463333379138258825621969416821633/ 7266156288540066197944574959711802353570870800 -64 the error is, -0.5291803209 10 `The `, 44, `-th apprx. is`, 120733610338615027996190010435213356517047676197/ 174181780904140382989045112088656766000859717200 -65 the error is, -0.1557984874 10 `The `, 45, `-th apprx. is`, 7188773397066071200052418897800041988750588170543/ 10371207730021727966396707913695458530740139026800 -67 the error is, -0.4586907569 10 `The `, 46, `-th apprx. is`, 65806198194682966271237463092990618996470212873791 /94938275795225371984549476540954522991737396543600 -68 the error is, -0.1350436369 10 `The `, 47, `-th apprx. is`, 69348341782161844309665586080818703072998198532037 /100048508782997791468237989834368854392193545685200 -70 the error is, -0.3975811775 10 `The `, 48, `-th apprx. is`, 91995840907026910246251856100014959188412871097024183/ 132721943458977760754444917451439064628142488726146800 -71 the error is, -0.1170510231 10 `The `, 49, `-th apprx. is`, 1120163803751964667069465351328462620606592977033703393/ 1616054764656277447220969490930415047735237293998042800 -73 the error is, -0.3446057042 10 `The `, 50, `-th apprx. is`, 214287449817218356335840894982030976809446910926444329/ 309151441176043530826653532749157424097881083033508400 -74 the error is, -0.1014536533 10 `The `, 51, `-th apprx. is`, 1827083538922494024488153994990786998947102154393958429773/ 2635924360893339481850468164186010894239049450495548604400 -76 the error is, -0.2986833535 10 `The `, 52, `-th apprx. is`, 94901301389915330563405461592579895060770702972780107478871/ 136913636889139737670255753277131362341882388732122149678800 -78 the error is, -0.8793313521 10 `The `, 53, `-th apprx. is`, 26085962257434443577703502700005875770385922463115591842221/ 37634088385617340767344013622044150149437668883194402418800 -79 the error is, -0.2588763701 10 `The `, 54, `-th apprx. is`, 167623772660653247077211586357724713677942800538925074447390623/ 241829985552623442817127350939411222396752295448260620217064400 -81 the error is, -0.7621328308 10 `The `, 55, `-th apprx. is`, 88956363277588681458981791740374806381494660745400959284161/ 128336904156094285706605506084340423531661217462210554730800 -82 the error is, -0.2243713628 10 `The `, 56, `-th apprx. is`, 207045653886592864312575569538597871498252041574760612118749809/ 298703738099800257801235009007666234702406223772564506773553200 -84 the error is, -0.6605455296 10 `The `, 57, `-th apprx. is`, 70351349739692931398686274414314793724235399883375495349800691/ 101495543389300058828912129156715809525548321041191264506286800 -85 the error is, -0.1944628928 10 `The `, 58, `-th apprx. is`, 901236320210986643590674525298706351095804336841433376252291517/ 1300209169837408294265792768479917393941419280615839004508951600 -87 the error is, -0.5724920350 10 `The `, 59, `-th apprx. is`, 39924131190694779685724260681165642474648592559631041813678520567/ 57598346080615744505651827922009032543875821967930686752061811600 -88 the error is, -0.1685392060 10 `The `, 60, `-th apprx. is`, 5833980127520020975216812382931278179063309282849287843697560753339/ 8416654198617896614271543892674757352169055315984211988272306597200 -90 the error is, -0.4961708978 10 `The `, 61, `-th apprx. is`, 7152620523559649915559997823561541093493158911989971389767656329451/ 10319050158700127989490942499396302169805441427188608887343794286800 -91 the error is, -0.1460698348 10 `The `, 62, `-th apprx. is`, 27743656532720879176300717575870854641249320235977055461224593218030383/ 40025635695883105536039317255114968534729459874526325608805647663584400 -93 the error is, -0.4300200922 10 `The `, 63, `-th apprx. is`, 725780568524608406463376041058250468845659807301771416306911031691173/ 1047080026984025033138275748043689486586108901667198744656357841476400 -94 the error is, -0.1265948293 10 `The `, 64, `-th apprx. is`, 8346993987349746739531663558999787802546656953643389812548475281698296947/ 12042166831879474582585737849900519735952935530492637682235672279602059600 -96 the error is, -0.3726852830 10 `The `, 65, `-th apprx. is`, 748367084176390601301564330039252170850444090567967162722335024631522607/ 1079665481105812158705840165128872921730238102949729477673240529672117600 -97 the error is, -0.1097154072 10 `The `, 66, `-th apprx. is`, 558343303276264313940276047892270973640626896051665704132014255363417247693/ 805519114750229040807702832599099328389795401010096724470077244785108666400 -99 the error is, -0.3229922759 10 `The `, 67, `-th apprx. is`, 2698081823077299817556705778515523191668380502378322283796040899930225327/ 3892509266066273994160638255067472172963742007844542303314476030470709600 -101 the error is, -0.9508583711 10 `The `, 68, `-th apprx. is`, 32096443972588568719089157199370757401284927126614972339362245668599740218037/ 46305380549423988249738331425216650167955784043741296253202015318591926797600 -102 the error is, -0.2799231061 10 `The `, 69, `-th apprx. is`, 15182687151633833587909785776084380360563521702888687584414540587575167076609/ 21903987461030712916659014059202137253338692386134431998120303012853687703200 -104 the error is, -0.8240639462 10 `The `, 70, `-th apprx. is`, 13517533887802960678662577499538491201118228670156\ 31429741808678382058096685241/1950167910498184085242610413429044808723642770893\ 172076798503646627405854736800 -105 the error is, -0.2425952885 10 `The `, 71, `-th apprx. is`, 245685457729644853154266017577147812489192459286494313472376972225879339149771/ 354449191485093672249315771959281260659305163074702458770011913545960660888800 -107 the error is, -0.7141725017 10 `The `, 72, `-th apprx. is`, 12940340186224832905128498348971030684598400691317\ 75872292092642197243946812487913/1866896461408273166140066487164037121708964416\ 571599582237480402651566898402026400 -108 the error is, -0.2102438152 10 `The `, 73, `-th apprx. is`, 91541944113409937493214444701233436431375070691801\ 49115311999385544925065506044877/1320671088057511528635792104817608222779617515\ 2293176450974787318195420878166605600 -110 the error is, -0.6189316669 10 `The `, 74, `-th apprx. is`, 22754115148791492655233515053133584792590292052998\ 6387552883269423452055069154740303/32827249084977924182031587088742120602240866\ 6034660646643360835763627658526064418400 -111 the error is, -0.1822055382 10 `The `, 75, `-th apprx. is`, 72235976926192765853213716039761881151970986052382\ 1679532898093348758130418208969007/10421448568518788956259401581417900781857331\ 96109538156311260481299587252881497429600 -113 the error is, -0.5363890002 10 `The `, 76, `-th apprx. is`, 29918438627971491035929190806958102690885642040774\ 997346910282235689349932086538523781/431631830397152725245498320178289514961338\ 91681902360426239349409162267596163448749600 -114 the error is, -0.1579056231 10 `The `, 77, `-th apprx. is`, 75066327056807967233476029810703015586085776184636\ 0978777248631073978042940309727226853/10829781778260587043863290708331390770411\ 71048560784964516896871988999880635781162784800 -116 the error is, -0.4648520956 10 `The `, 78, `-th apprx. is`, 29445862352135186969544483577011903787736249962643\ 79385329982422697966825207932534691733/4248139959012446177963987966286268343130\ 534454212555319761415001503227317450032767712800 -117 the error is, -0.1368458010 10 `The `, 79, `-th apprx. is`, 17620678663288156917279087894967346815382392017198\ 9158704482417037369954230503703669467991/25421265724623792619811193344649304392\ 5859318683552118148448611131037645188265659001525600 -119 the error is, -0.4028539849 10 `The `, 80, `-th apprx. is`, 47423564705003013535670128593522967091270308620010\ 47292072661917170981598908802988550463939/6841774162118472448227749785439499494\ 383325195873984003540682079500419392106376410526002400 -120 the error is, -0.1185941798 10 `The `, 81, `-th apprx. is`, 71968123514462821160526484982512487669272147893464\ 21279831107671040423512553155260151843/1038280548963999086534153908448463740748\ 9846339842117256320400413166795406148498130312800 -122 the error is, -0.3491231352 10 `The `, 82, `-th apprx. is`, 21908761991098303634396151089459415519419932250424\ 8716150134269399363395828972445971132039228187/31607662276574134516401656413399\ 4160751247744866443835129091721195502502627570569105012886095200 -123 the error is, -0.1027764093 10 `The `, 83, `-th apprx. is`, 38459047337935769859290283219728409156846101905328\ 252093534541801684301849471711193018236747189/554846768717538247799276121873232\ 07778632876276254635707972454591874171656069504305937439754400 -125 the error is, -0.3025574329 10 `The `, 84, `-th apprx. is`, 61026335343248019167130193250090684285115673017818\ 8603223575730239765272253144661768320070048620981/88042391363330793747666818682\ 0937077882365718893571079809144524620233388786434566170877021728317600 -127 the error is, -0.8906802095 10 `The `, 85, `-th apprx. is`, 39285552331350300517833438236326842713748624279156\ 2012392623781376392721255650959177832233077993923/56677071527022933520081528839\ 3465581921024025094793929717510678472465647116703498348162021355080800 -128 the error is, -0.2622016274 10 `The `, 86, `-th apprx. is`, 11593845542069881664025769329162206852617281271167\ 013681392602637107358462134366491160888136427991/167263834683768340475600388378\ 20331807074229542398113989127687911060695383571074665630078455461600 -130 the error is, -0.7718778549 10 `The `, 87, `-th apprx. is`, 13189858251977775068358756262360587159093879322248\ 693952590796785928170414430010780152532004665829559/190289430901567076156294215\ 37127690986202584687764397195871665012139099980615991533581419769075058400 -131 the error is, -0.2272277725 10 `The `, 88, `-th apprx. is`, 98272584464243290618842473797517496457126573099662\ 672774219752225173030540524988100759747308384596883/141777370261905584264482281\ 408950067872718651899549420531856692218983794117535416757012042297373960800 -133 the error is, -0.6689195440 10 `The `, 89, `-th apprx. is`, 17325673071727484845804254328772493171662994776576\ 36971476724524634196247688409351810638846739856431/2499566262064469598736392018\ 857021242691144274822827019133014451228025997081902805159935422166565600 -134 the error is, -0.1969182716 10 `The `, 90, `-th apprx. is`, 20563685334602125866421788232185716326250268880565\ 16032067458435287618048696413494062076668409040611265233/2966712685463159114386\ 7635114441798801689688108809271165666864575514962148641765696090496387460146537\ 60800 -136 the error is, -0.5796926753 10 `The `, 91, `-th apprx. is`, 48994110240118604384222016773988277989836047320735\ 3720119081086039447749954204682646518827806227673609/70683559876186290736462969\ 9952590564247202614903080838620472487323585473920382337984176124679242426400 -137 the error is, -0.1706511775 10 `The `, 92, `-th apprx. is`, 76762316368306402755612766631264252344086862648306\ 69318197628340425455231929348897972216109155124516986941/1107446131517053507713\ 7504942497516455717502842420600714526599986499382658275242668693256515515161106\ 813600 -139 the error is, -0.5023662502 10 `The `, 93, `-th apprx. is`, 18775173756370824511654460038235760716705326264519\ 3773577859439869619970690227800246841610176764911037663/27086850070144799359912\ 3562293372682779734539153061561601408453194997100973054944977601177449188045784\ 800 -140 the error is, -0.1478874498 10 `The `, 94, `-th apprx. is`, 54116200067344789343679351586478178969439999471398\ 92043828008034272799675930885090650770880342544804370901/7807317346891331516120\ 9265309156337278381169289409558401770389560672232059394224818653262522787175232\ 29600 -142 the error is, -0.4353533533 10 `The `, 95, `-th apprx. is`, 30089878537703884531298256462257321452675731329360\ 23313938421228013242055740495895882778119382976908461431/4341051854729664815396\ 0232442892832359041687533571101126528865749244977235215332964876360666130320555\ 57600 -143 the error is, -0.1281599077 10 `The `, 96, `-th apprx. is`, 29225858172482557318448508899346556005228258741280\ 4361050320923347370292247544062862576618760312955910374303/42164000651164768393\ 8693597661012327760387532090244410349231530944292133369331279390836706365857710\ 198104800 -145 the error is, -0.3772786291 10 `The `, 97, `-th apprx. is`, 98570961381220200730158033290714320099876291624973\ 1820409886196425919602584602534901314251102919257105279067/14220783716034391043\ 7396430334159110531364630929581470129854590581841540341691038067221597929495116\ 5676967200 -146 the error is, -0.1110636546 10 `The `, 98, `-th apprx. is`, 28274666745508209841178952116961641667263569344986\ 664104612215504567669352399722762068290310588731444629039554499/407917214965327\ 8070920139254723528063934507817580309623397015065278264445166209114716476271287\ 0650212630961978400 -148 the error is, -0.3269500879 10 `The `, 99, `-th apprx. is`, 54651967403224467054652918202224733149867764260505\ 164760503668332793677607405418462635170328144259058886755209423/788461223474572\ 1778808513096427022655718425077725316649701401085733921090868737456711699761707\ 7668037341935216800 -150 the error is, -0.9624777715 10 `The `, 100, `-th apprx. is`, 8557001896805197774860371515007473726483974203460\ 633094374636984963232618308603100324496776487461420893870100906007/123451442013\ 9831223897938188454745947120289626971283279711606968055361107134907547046148091\ 1747034700851765169591200 -151 the error is, -0.2833346682 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - ln(2) | <= ------------- | bn | 1.291683540 bn since , 1.291683540, is larger than 1, this sequence probably suffices to \ give an Apery-style proof for the irrationality of, ln(2) This ends this article, that took, 0.412, seconds. to generate. ---------------------------------- if the input is: DelSeqIntV(x^n*(1-x)^n/(1+2*x)^(n+1),x,n,0,1,log(3),A,100): the output is: Better and Better Rational Approximations to, ln(3) By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / n n | x (1 - x) A(n) = | ---------------- dx | (n + 1) / (1 + 2 x) 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: A(0) = 1/2 ln(3), A(1) = - 1/2 + 1/2 ln(3) Note that these are linear combinations of, ln(3), and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients (n + 1) A(n) + (-8 n - 12) A(n + 1) + (4 n + 8) A(n + 2) = 0 and in Maple input form : (n+1)*A(n)+(-8*n-12)*A(n+1)+(4*n+8)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , ln(3), and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , ln(3), yields the following sequence of terrific approximations to, ln(3) `The `, 1, `-th apprx. is`, 0 the error is, -1.098612289 `The `, 2, `-th apprx. is`, 1 the error is, -0.09861228867 `The `, 3, `-th apprx. is`, 12/11 the error is, -0.007703197759 `The `, 4, `-th apprx. is`, 56/51 the error is, -0.0005730729818 `The `, 5, `-th apprx. is`, 1460/1329 the error is, -0.00004193501875 `The `, 6, `-th apprx. is`, 12244/11145 -5 the error is, -0.3046855638 10 `The `, 7, `-th apprx. is`, 55804/50795 -6 the error is, -0.2205511690 10 `The `, 8, `-th apprx. is`, 676476/615755 -7 the error is, -0.1592976408 10 `The `, 9, `-th apprx. is`, 37854164/34456345 -8 the error is, -0.1148931438 10 `The `, 10, `-th apprx. is`, 600226324/546349545 -10 the error is, -0.8278653532 10 `The `, 11, `-th apprx. is`, 8510202668/7746320295 -11 the error is, -0.5961104109 10 `The `, 12, `-th apprx. is`, 166709729548/151745735295 -12 the error is, -0.4290159348 10 `The `, 13, `-th apprx. is`, 4769186865196/4341100963815 -13 the error is, -0.3086405296 10 `The `, 14, `-th apprx. is`, 10111600419236/9203975345565 -14 the error is, -0.2219742460 10 `The `, 15, `-th apprx. is`, 1600941245404892/1457239521092355 -15 the error is, -0.1596060256 10 `The `, 16, `-th apprx. is`, 2887528212507812/2628341447016285 -16 the error is, -0.1147395157 10 `The `, 17, `-th apprx. is`, 334039379669895308/304055746613634015 -18 the error is, -0.8247245005 10 `The `, 18, `-th apprx. is`, 10284092306704545292/9360984227813753535 -19 the error is, -0.5927186450 10 `The `, 19, `-th apprx. is`, 49749464526445799884/45283914115651306095 -20 the error is, -0.4259329302 10 `The `, 20, `-th apprx. is`, 156120098163458186564/142106637413212116645 -21 the error is, -0.3060511640 10 `The `, 21, `-th apprx. is`, 588100074716346622084/535311757188992636853 -22 the error is, -0.2198937451 10 `The `, 22, `-th apprx. is`, 867185067574886518724/789345865251706429381 -23 the error is, -0.1579801371 10 `The `, 23, `-th apprx. is`, 12650739122972962604100/11515198995552784638821 -24 the error is, -0.1134923969 10 `The `, 24, `-th apprx. is`, 531127117560488310360644/483452736728800225854181 -26 the error is, -0.8152840742 10 `The `, 25, `-th apprx. is`, 8469059736847191375475652/ 7708870385124274489695429 -27 the error is, -0.5856413939 10 `The `, 26, `-th apprx. is`, 4258893978445794884613645484/ 3876612361226195019853827975 -28 the error is, -0.4206660572 10 `The `, 27, `-th apprx. is`, 62353045070162901211906813172/ 56756187522493415148732679425 -29 the error is, -0.3021537667 10 `The `, 28, `-th apprx. is`, 1027749633946484143410383844316/ 935498031969462953950144318275 -30 the error is, -0.2170226419 10 `The `, 29, `-th apprx. is`, 210953129516176926555791090148724/ 192017813465315949987889708920225 -31 the error is, -0.1558726769 10 `The `, 30, `-th apprx. is`, 11218572669408041245246351204533524/ 10211584910458950193412800262925225 -32 the error is, -0.1119499970 10 `The `, 31, `-th apprx. is`, 80057415914995719936629997648836/ 72871400348208767840077965219525 -34 the error is, -0.8040228505 10 `The `, 32, `-th apprx. is`, 9372250016601435475773741142710983332/ 8530989606864654550664694826074824925 -35 the error is, -0.5774359816 10 `The `, 33, `-th apprx. is`, 2203551345723364245625811131100894718092/ 2005758872763761863302897296318098910175 -36 the error is, -0.4146973466 10 `The `, 34, `-th apprx. is`, 4049504650152085031469254927012791204364/ 3686017980976215601660838689659872982975 -37 the error is, -0.2978182819 10 `The `, 35, `-th apprx. is`, 2589642076296388154291837731747055030876/ 2357193800768341789657568179156902034275 -38 the error is, -0.2138773386 10 `The `, 36, `-th apprx. is`, 9959469243823444723951265312326510472996/ 9065499582111625348762658881623719910525 -39 the error is, -0.1535932299 10 `The `, 37, `-th apprx. is`, 3225413338476205428157647051294626601976036/ 2935897742766466938762244884663466095876525 -40 the error is, -0.1102995556 10 `The `, 38, `-th apprx. is`, 12922316142738038812515848852340149962060292/ 11762399051993368939835519112467120744747925 -42 the error is, -0.7920822635 10 `The `, 39, `-th apprx. is`, 3236222182504740669088579491248022157284331364/ 2945736376595730965999770222446212142787339725 -43 the error is, -0.5688032298 10 `The `, 40, `-th apprx. is`, 192302295778699326198552641942824013531771396/ 175041092988168611837646663717990748546245525 -44 the error is, -0.4084598936 10 `The `, 41, `-th apprx. is`, 351516124866090742669313416062646234484628734404/ 319963765644973242777519135821207019838062982725 -45 the error is, -0.2933139461 10 `The `, 42, `-th apprx. is`, 2415015212729877530938613135727411889001795930764/ 2198241579527290912322690463403209125573122362975 -46 the error is, -0.2106261197 10 `The `, 43, `-th apprx. is`, 55977802387684450371161219859014965295686379840932 /50953191553635825787030312279136535552688836704925 -47 the error is, -0.1512475086 10 `The `, 44, `-th apprx. is`, 4439331194872834447464806754100253680785105223618372/ 4040853393561442791470981402557381917402898968005925 -48 the error is, -0.1086077864 10 `The `, 45, `-th apprx. is`, 131034936290992530816497786809779055461953863772119108/ 119273139070609949959243861880531404819935198345332325 -50 the error is, -0.7798851414 10 `The `, 46, `-th apprx. is`, 241795267546682586220322156385526473073238119030261492/ 220091537333721591446084688455937684700251609652567425 -51 the error is, -0.5600121918 10 `The `, 47, `-th apprx. is`, 324572748039121729238952033667555647046919966234268076/ 295438847159277515961299813491452189778634395209464775 -52 the error is, -0.4021255368 10 `The `, 48, `-th apprx. is`, 18223103331160395945834017179672999494485954029816010852/ 16587383482896383083257053991988783714771755188896838925 -53 the error is, -0.2887508886 10 `The `, 49, `-th apprx. is`, 36611484771564512970436818504866547355595384924361650041732/ 33325209584129116785832314873406248679331500450359598295925 -54 the error is, -0.2073397944 10 `The `, 50, `-th apprx. is`, 3313393185750127628663562466674244412092743721849802216364292/ 3015980450907829150690730445801000561715095507938643905699925 -55 the error is, -0.1488811787 10 `The `, 51, `-th apprx. is`, 48967981064820965812350077422912648279517679785965174237202444/ 44572577213920253370829834741045912774404115542495583645946175 -56 the error is, -0.1069042237 10 `The `, 52, `-th apprx. is`, 66091275569284933673942645598572781549750229682428885780676/ 60158871560967110319321964855777612598910126455523418520325 -58 the error is, -0.7676230054 10 `The `, 53, `-th apprx. is`, 5932110343252050710185343435630780107400157066047548303429268/ 5399639531106809547548048374873135549623651869655242697465225 -59 the error is, -0.5511873764 10 `The `, 54, `-th apprx. is`, 8454692335606317353138852328219703074002579369682279349195933348/ 7695792612930143855693234279209837006763328321260026240902133725 -60 the error is, -0.3957754117 10 `The `, 55, `-th apprx. is`, 76007051947336723987502976536891246136372815901531264639794876276/ 69184600182729634742731052578367982408478700205184064673799137825 -61 the error is, -0.2841820990 10 `The `, 56, `-th apprx. is`, 2389193283852332716049819205901205281092944532526763811342495347692/ 2174737446955781320642308926765776241879330128612361501012782202775 -62 the error is, -0.2040530432 10 `The `, 57, `-th apprx. is`, 141390746351382269856829206336877404238251255653421628542551389293884/ 128699403611073528268675947046175798263213570344577343711381872023675 -63 the error is, -0.1465169829 10 `The `, 58, `-th apprx. is`, 1033686689310058348036292997144678365785503969091231803469029129356/ 940902172652043454872243784552166801704541540206566943133792740575 -64 the error is, -0.1052038096 10 `The `, 59, `-th apprx. is`, 3870385667581924178021534974017091592018504494848018084855178651196548/ 3522976856807366419541972505674108685564530032399977663805587325612725 -66 the error is, -0.7553942019 10 `The `, 60, `-th apprx. is`, 11418878004032798281901363921075613394245731227360233647563427981125844/ 10393910683337018912778923583103614186881760234292298919409815344494425 -67 the error is, -0.5423936222 10 `The `, 61, `-th apprx. is`, 12509097793211531018476923875562013817842626948646832269833747781251545396/ 11386271501092342305695786390224618714994095480238109754729227891118527825 -68 the error is, -0.3894523335 10 `The `, 62, `-th apprx. is`, 1412201196302784004492423677323852225018055767487307829859520241059039498196/ 1285440924764140691043713311799218770534775522228295665917877414566231372825 -69 the error is, -0.2796358921 10 `The `, 63, `-th apprx. is`, 444925625046279768902209893961818147764372699195498002343742335694585060188/ 404988756848587930181203107925319031478803382055151311759950741924319693475 -70 the error is, -0.2007846180 10 `The `, 64, `-th apprx. is`, 1173076045601418194200780467069516085720755503898568827737809385432595637724/ 1067779832522703532527728623568131881862269218286534532956177878498877108675 -71 the error is, -0.1441673568 10 `The `, 65, `-th apprx. is`, 21162655193425767257050104099114676038630262635568325540088599171165359755004/ 19263078896634294299917301318257546961165333749815642302602251220665819189675 -72 the error is, -0.1035148015 10 `The `, 66, `-th apprx. is`, 213675133223654145414274184671933271593016099080687613166179485059554353548044/ 194495488014885397803671282897670205570397214986287042312464724895209578578175 -74 the error is, -0.7432536258 10 `The `, 67, `-th apprx. is`, 55925710324028483722439762133404662922177511313734\ 397955359316400440426502409372/509057753139002227210061854885922114084402086879\ 68453519670576268783363664596275 -75 the error is, -0.5336674648 10 `The `, 68, `-th apprx. is`, 69398290624182268679784009650878717792946954616705\ 11414159258522099695972346407868/6316904638698016874215661238265397197932381450\ 151782935443518905292558630147000475 -76 the error is, -0.3831806448 10 `The `, 69, `-th apprx. is`, 18697631978937318958707510315136883577334121395800\ 654819381773425734004013356922876/170193180722611284079243866417380965233743202\ 47495574317057213860895257437374372075 -77 the error is, -0.2751285033 10 `The `, 70, `-th apprx. is`, 11430303429252513896818301296698861924814940772998\ 78596006826705850870727984811297132/1040431055355289561780669165751787974908151\ 863414545458146286267223025087422747590775 -78 the error is, -0.1975453570 10 `The `, 71, `-th apprx. is`, 13329220246159368267823323617149377073488162584252\ 993331596343719516416716222128084/121327791284028865227651619146848705915524133\ 72034165494197951862756961839319910425 -79 the error is, -0.1418395422 10 `The `, 72, `-th apprx. is`, 13110109656029007953467231944701017339820458435674\ 0010853397447970537332600765306733132/11933336074296877399893963227572588394679\ 7546584563439308248769050900845643951517909775 -80 the error is, -0.1018420421 10 `The `, 73, `-th apprx. is`, 39647508976389011240444395716624312121114918780719\ 4695060213067553470157995583722460856748/36088717908348927056464285497626666653\ 2649122731929229906571632231206400838165909850539975 -82 the error is, -0.7312336876 10 `The `, 74, `-th apprx. is`, 53637662203208984922233839733029881721870974361979\ 636224901491363670668226879558106231742828/488231041619204935268803633338269430\ 80412998067661602743333770833885664879897337138210745975 -83 the error is, -0.5250306108 10 `The `, 75, `-th apprx. is`, 72300072781658753624742545246394744126599309344028\ 220074364131412397814362276939826708499196/658103623338411200760851888602499200\ 62236746163996490287243579907038506239098398289126268075 -84 the error is, -0.3769748932 10 `The `, 76, `-th apprx. is`, 14741526037741835656328323117113675083821248507905\ 23553874631875507956784562583063683372429916/1341831525989351503503004410247243\ 085414732043317017738330384391789613777648766251992303506275 -85 the error is, -0.2706696880 10 `The `, 77, `-th apprx. is`, 71677360883066298697309587268799854284586624647856\ 1643640144300596702071636688261525213490796/65243545536854994310012457547854067\ 2596335494890579473222240051093718683393096322359639470775 -86 the error is, -0.1943418063 10 `The `, 78, `-th apprx. is`, 81058515580119816278798996844634216595224723684988\ 268307994528146942877559456964850065873907852/737826405331677111933577834055332\ 23368513254969420841962736747295501738749383675791907206686175 -87 the error is, -0.1395379337 10 `The `, 79, `-th apprx. is`, 25580793245497360634502316819234961883956516598463\ 132739104664305074672371187012243675348639716/232846414602825432935845829680348\ 57037257626983927916110841201201977022023306331459279455292525 -88 the error is, -0.1001884831 10 `The `, 80, `-th apprx. is`, 26246728007157426644187772325092674508622962605842\ 2185731454801067396301829677610774173057053412/23890801402719929063665696499234\ 8773818831537614268914509255277334262554791465821780157341818925 -90 the error is, -0.7193542210 10 `The `, 81, `-th apprx. is`, 12448594310173048641671626989641067345024859696713\ 594158076160955056955063847852974763840160267292/113311988574831641332054669188\ 81480977922763992352629068644077201797288492383134240816767335638675 -91 the error is, -0.5164963922 10 `The `, 82, `-th apprx. is`, 20520677567475781894159690178215867456340615340698\ 460211346129762544285622149112823159900346674899396/186787256788778465266794554\ 11454212690284873258971418639954830931475750358909949672109844389941651525 -92 the error is, -0.3708440432 10 `The `, 83, `-th apprx. is`, 51759566526731352813389685254796808591554892046649\ 16198912748877182920593910891929791955316120460046412/4711358780583228667101190\ 027342678796962655448309158960948795447467613464367740422027169740248720684175 -93 the error is, -0.2662654759 10 `The `, 84, `-th apprx. is`, 79682119863294484150001259048728150329237808319984\ 3865751370554957208429680745441864702824993346288141932/72529791160352129395531\ 4032824407656518323922324998913388357180195201829620036229948640760981881588437\ 175 -94 the error is, -0.1911780223 10 `The `, 85, `-th apprx. is`, 76285621398854366385386214481332491975512502987182\ 6566248242094771892612988150262152511070541937743808108/69438165024840915833254\ 9425235546237053709134761440058466123417101060573010990534313896860207870144827\ 575 -95 the error is, -0.1372652606 10 `The `, 86, `-th apprx. is`, 39880981629361612707875717072441525876907604230410\ 18012494203024646335151948864978556647432482831536893036/3630123387542922380465\ 4983142032361105178880418331388571182166520287363927831521240724731068867587968\ 30775 -97 the error is, -0.9855596115 10 `The `, 87, `-th apprx. is`, 65107731288539220574769215752841611409838434812205\ 4802882378270644536945303243817625310432816809733224391588/59263610975507883391\ 6654455362081026997675954805956795509284346181463209521313407525678358827168264\ 086839325 -98 the error is, -0.7076275862 10 `The `, 88, `-th apprx. is`, 12079413035045705111496627308311500119897652126979\ 52584126579410331885349769815263330454344081040995172622076/1099515557912614127\ 8590861835306608614884838621411750219015922526868363907131206400367498501019581\ 84407119275 -99 the error is, -0.5080731273 10 `The `, 89, `-th apprx. is`, 11402156181994077697712309493811718549881141936724\ 2092937973504507334156756393766757900857523004227464996876/10378689824976702282\ 8531780727700759313938559825221025310277039855903768502793782565542835359648372\ 613011775 -100 the error is, -0.3647936927 10 `The `, 90, `-th apprx. is`, 11843971864432248883344682966240271004698377042673\ 805313838292272441386387921637289072736312375576795758971254204/107808477900708\ 3160655613207055152339021082904207394519545035273238580143965950326378370859462\ 3953706301827976475 -101 the error is, -0.2619196336 10 `The `, 91, `-th apprx. is`, 14531116990420464440541193413792467112945161319629\ 49498601380244958604429483397764428703202225221545679802074884/1322679269138442\ 0467497504526371424618832476918385137958769203798385318240313715975847177585404\ 69518825885536725 -102 the error is, -0.1880565387 10 `The `, 92, `-th apprx. is`, 32629323500134125683766181056729267520017021910372\ 8263872285012481307099563626370623296003249527178557106927651404/29700490188118\ 9775567980009785788822499638005180021763827685381835321332076619327156283912997\ 390233677707583560975 -103 the error is, -0.1350232272 10 `The `, 93, `-th apprx. is`, 96889754392534707221603711855361457699330731849950\ 02270670123098220765369892730891831661887780520881057476973664348/8819285510632\ 5013190869506490960801053016773676931328571941273565675278767685180355388911771\ 65240920882179370511075 -105 the error is, -0.9694562218 10 `The `, 94, `-th apprx. is`, 10475297829245671374414550920918495322457913790420\ 4722118545957635642585397469050363853632357091593576399437668/95350269947783684\ 6431472150903812546809926975473446665711031403853581247037966869078539457763990\ 53930418725325 -106 the error is, -0.6960614848 10 `The `, 95, `-th apprx. is`, 26701983337231361485573666750098249277805716361729\ 9927419301093109042231696927893398527196825709655717713123755985692/24305192662\ 2841731009957617305927292085186810965528324845204151715704430983371784671276265\ 448984270103844775025038675 -107 the error is, -0.4997659898 10 `The `, 96, `-th apprx. is`, 49564225428163593549695557323532650844625773129209\ 1339160485410132996472818137218865105766704845205504058790356259108/45115302222\ 0898578779521904335907454534776592330854423400067482818609285215518464450589771\ 238460767664262131700035325 -108 the error is, -0.3588273150 10 `The `, 97, `-th apprx. is`, 28935610000065525086681332886093047963884150381942\ 2022943306510643939634088328161621687733058197641041702022174744732/26338327268\ 4354258592636168115127110988105298525018500096115612170700714955967065762882342\ 349328845893227694505690675 -109 the error is, -0.2576344915 10 `The `, 98, `-th apprx. is`, 21206598005998285016992406274376027072163735215990\ 114730318658537663537032774947887191588204137198430158266939873945463556/193030\ 7736836611093269947235149144199489582561279834341236497014440194413632409625994\ 7959096637578695494315362533999693525 -110 the error is, -0.1849789182 10 `The `, 99, `-th apprx. is`, 18527092396040639921133130039896201319253921448657\ 207888837947514054604901068028721754100393669591911725052655652236465708/168640\ 8625421599185514124388463945467490144213346602033758836198331748902081077139880\ 1780782709752091602339227000809161575 -111 the error is, -0.1328128802 10 `The `, 100, `-th apprx. is`, 5847549416683276690155994027660068374953261699921\ 10371036119486375624429851039781851160918488773458788446272965293400129284/5322\ 6688587039998969756494130065869772975098873605279597737262761441908767652273280\ 4417816751050447782417659193626726688725 -113 the error is, -0.9535816654 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - ln(3) | <= -------------- | bn | 0.9067162948 bn since , 0.9067162948, is smaller than 1, this sequence does not provide an \ Apery-style proof for the irrationality of, ln(3) but this sequence of rational approximations is still neat! since it is effe\ ctive! This ends this article, that took, 0.422, seconds. to generate. ---------------------------------- if the input is: DelSeqIntV(x^n*(1-x)^n/(1+4*x)^(n+1),x,n,0,1,log(5),A, 100); the output is: Better and Better Rational Approximations to, ln(5) By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / n n | x (1 - x) A(n) = | ---------------- dx | (n + 1) / (1 + 4 x) 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: A(0) = 1/4 ln(5), A(1) = - 1/8 + 3/32 ln(5) Note that these are linear combinations of, ln(5), and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients (-n - 1) A(n) + (12 n + 18) A(n + 1) + (-16 n - 32) A(n + 2) = 0 and in Maple input form : (-n-1)*A(n)+(12*n+18)*A(n+1)+(-16*n-32)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , ln(5), and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , ln(5), yields the following sequence of terrific approximations to, ln(5) `The `, 1, `-th apprx. is`, 0 the error is, -1.609437912 `The `, 2, `-th apprx. is`, 4/3 the error is, -0.2761045791 `The `, 3, `-th apprx. is`, 36/23 the error is, -0.04422052113 `The `, 4, `-th apprx. is`, 476/297 the error is, -0.006744309740 `The `, 5, `-th apprx. is`, 2900/1803 the error is, -0.001007518646 `The `, 6, `-th apprx. is`, 204436/127035 the error is, -0.0001491337511 `The `, 7, `-th apprx. is`, 652932/405695 the error is, -0.00002197188763 `The `, 8, `-th apprx. is`, 22193372/13789545 -5 the error is, -0.3228403554 10 `The `, 9, `-th apprx. is`, 870727068/541013305 -6 the error is, -0.4735526296 10 `The `, 10, `-th apprx. is`, 38725577236/24061555035 -7 the error is, -0.6938240752 10 `The `, 11, `-th apprx. is`, 128349488828/79748021745 -7 the error is, -0.1015726817 10 `The `, 12, `-th apprx. is`, 21160007209612/13147451707005 -8 the error is, -0.1486088492 10 `The `, 13, `-th apprx. is`, 47173223692036/29310371859615 -9 the error is, -0.2173280557 10 `The `, 14, `-th apprx. is`, 9259734869292932/5753396759165055 -10 the error is, -0.3177126115 10 `The `, 15, `-th apprx. is`, 2832775873442764/1760102611952685 -11 the error is, -0.4643361315 10 `The `, 16, `-th apprx. is`, 1419161845458673412/881774832377934495 -12 the error is, -0.6784742912 10 `The `, 17, `-th apprx. is`, 38385120979802203052/23850016632049653045 -13 the error is, -0.9911855882 10 `The `, 18, `-th apprx. is`, 9947159201251572732556/6180517511363742031485 -13 the error is, -0.1447807781 10 `The `, 19, `-th apprx. is`, 1022819434987049491284/635513446703978357515 -14 the error is, -0.2114520400 10 `The `, 20, `-th apprx. is`, 1089707982899162265136868/677073638243737702022055 -15 the error is, -0.3087922496 10 `The `, 21, `-th apprx. is`, 4449974803754707636058340/ 2764924803482852756232487 -16 the error is, -0.4509010732 10 `The `, 22, `-th apprx. is`, 9747052856250666413600332/ 6056184448587585226269069 -17 the error is, -0.6583579174 10 `The `, 23, `-th apprx. is`, 33247597743065651286555836/ 20657893967951993786798193 -18 the error is, -0.9611990487 10 `The `, 24, `-th apprx. is`, 11749795783241471482261455148/ 7300558594069142483829912021 -18 the error is, -0.1403262301 10 `The `, 25, `-th apprx. is`, 160629494996007413847978747660/ 99804716761687759295713629533 -19 the error is, -0.2048527512 10 `The `, 26, `-th apprx. is`, 81441059462068273265588709044/ 50602175351329644824543265075 -20 the error is, -0.2990369295 10 `The `, 27, `-th apprx. is`, 211609864094398635016748189941092/ 131480601059261524867227782611975 -21 the error is, -0.4365059717 10 `The `, 28, `-th apprx. is`, 29360214077877502155334621975902172/ 18242526692734211224642452653997225 -22 the error is, -0.6371472690 10 `The `, 29, `-th apprx. is`, 1409126986383507145307124968051878036/ 875539823870779415184389491271007675 -23 the error is, -0.9299838204 10 `The `, 30, `-th apprx. is`, 57346228424228665343947794008562201916/ 35631215085208668508101188863553785425 -23 the error is, -0.1357370242 10 `The `, 31, `-th apprx. is`, 721719979225117461586865553756230029564/ 448429836062202753535053007195794153825 -24 the error is, -0.1981115810 10 `The `, 32, `-th apprx. is`, 345759587581268260908515130284635548544556/ 214832510723165683403615054750563659275925 -25 the error is, -0.2891419459 10 `The `, 33, `-th apprx. is`, 19008562417478548914896542019293091324663012/ 11810683885736330669839004689624877807079975 -26 the error is, -0.4219908005 10 `The `, 34, `-th apprx. is`, 882161613044734210775745373421871928917214604/ 548117828112151527512237316288385888067963325 -27 the error is, -0.6158661778 10 `The `, 35, `-th apprx. is`, 3033966903333863926569171139977962063110989748/ 1885109627338974435180214960837082522879404275 -28 the error is, -0.8987977015 10 `The `, 36, `-th apprx. is`, 46975585377776051230527215324444722186469778892/ 29187572266599940250388141161411065290103543725 -28 the error is, -0.1311687675 10 `The `, 37, `-th apprx. is`, 3266566998154162387932377472995129726603609132/ 2029632192032704081764677809139162830969960725 -29 the error is, -0.1914222400 10 `The `, 38, `-th apprx. is`, 20601042231482464298833550216571340314230217347876 /12800147226757956222888485492318421600853279361175 -30 the error is, -0.2793497867 10 `The `, 39, `-th apprx. is`, 70963854613703720884895872462079783650293226390292 /44092321962503408086708383126977631988299441992475 -31 the error is, -0.4076606165 10 `The `, 40, `-th apprx. is`, 3301171877609693032799033873106430970317478447286004/ 2051133412544649470179293070563899907787085874999075 -32 the error is, -0.5949001155 10 `The `, 41, `-th apprx. is`, 4137845352217316513036225593875540143758973952453412/ 2570987871137740239265175077656773835357375368589975 -33 the error is, -0.8681296964 10 `The `, 42, `-th apprx. is`, 28955881876090254036054729679711724770796839712197825708/ 17991300970596387656837943817407877503763323160724193525 -33 the error is, -0.1266837112 10 `The `, 43, `-th apprx. is`, 4755751932163468223179889282518509534655666545789467956/ 2954914815552535955692834302496707965655290046661494675 -34 the error is, -0.1848642183 10 `The `, 44, `-th apprx. is`, 6001618370676009690275500238660249901094206941651753828/ 3729015157595742539878281928792921651542691332938480775 -35 the error is, -0.2697622267 10 `The `, 45, `-th apprx. is`, 21912630998485543908810080577304889516340488754816341947588/ 13615083147472936634955815155233725550451159820974480553775 -36 the error is, -0.3936460261 10 `The `, 46, `-th apprx. is`, 3063383744530648769228646769108064175084566264045142069281908/ 1903387338438929238173659539309773891226091216230588333454275 -37 the error is, -0.5744170123 10 `The `, 47, `-th apprx. is`, 10576932979771411168153675019721735605053214706317628449980916/ 6571817960827669709790199892422208204775800946391516515096675 -38 the error is, -0.8381960662 10 `The `, 48, `-th apprx. is`, 17129861831666058538067591974446397084253703600929584769707948/ 10643381580193422588573149888007519901166616018973043938857525 -38 the error is, -0.1223097394 10 `The `, 49, `-th apprx. is`, 3094063528350106710056522249617848440911109785685050697942539412/ 1922449759910695175129035752184732166512486406621181776004620475 -39 the error is, -0.1784735002 10 `The `, 50, `-th apprx. is`, 54214720275194446812940382654070045958015790974974428407302645476396/ 33685499674355601227870702494238052300567843282200501836377836891925 -40 the error is, -0.2604257264 10 `The `, 51, `-th apprx. is`, 187352137053597538523331702799613950881353057749717047742121539454708/ 116408427815800449717383863262319092801237613504259343877447729053475 -41 the error is, -0.3800070221 10 `The `, 52, `-th apprx. is`, 4249970101279487082631255248454947609824233266736338615192525413188/ 2640654894758796885869016863729384152709481350982056760274781976975 -42 the error is, -0.5544943111 10 `The `, 53, `-th apprx. is`, 3555592341408979701845645604669839217654669932641936495571389908014812/ 2209213734770005338366133699631986719567208614702772078747422271778025 -43 the error is, -0.8090967043 10 `The `, 54, `-th apprx. is`, 2932181772595477294661481368040541574907437874917749307246301544677535004/ 1821866969792497444661790680446862888569443234328053996245931195644206425 -43 the error is, -0.1180597213 10 `The `, 55, `-th apprx. is`, 1126719399669320060077534289573117839775835313798495380292650733638032172/ 700070124460582084761027673081272947719643086741981195561763791309597525 -44 the error is, -0.1722666351 10 `The `, 56, `-th apprx. is`, 5845860642369818598741272923668379425603257120244072735268290276087199276/ 3632237439671461501859407240189571776131130865734163400159306556871474325 -45 the error is, -0.2513615154 10 `The `, 57, `-th apprx. is`, 242684760854557025737683762839354917710742978857943891025708214362381468724/ 150788520004181231638822360415035336136192157029462935604208721335727070675 -46 the error is, -0.3667707516 10 `The `, 58, `-th apprx. is`, 3778650087293058778743707080175135851224254372282400037690226762847318778676/ 2347807304711904132327827495067224543544323932715384966144277851731462893075 -47 the error is, -0.5351665834 10 `The `, 59, `-th apprx. is`, 1188754340325588582091957417845666776340877284115563627525124153143632240372/ 738614600253654089040904902063018074823026459120851792441480766409318058275 -48 the error is, -0.7808754082 10 `The `, 60, `-th apprx. is`, 12016047883487313273034062685705095235388327553171\ 539880135247847901536326923212/746599032535175179596068952288474617247054917121\ 6887687558726794115732286612525 -48 the error is, -0.1139391690 10 `The `, 61, `-th apprx. is`, 83189021890224121993568059576799508458649123365542\ 470552655758540546102295968348/516882454722405192550307263202469796074534217860\ 01566058798768559563690068617225 -49 the error is, -0.1662505020 10 `The `, 62, `-th apprx. is`, 26352479623435416036294305566952591095165366964677\ 024048970761038889743508821082484/163737162023107479148687330069265348223150163\ 73529164419325160413004585163192044675 -50 the error is, -0.2425780865 10 `The `, 63, `-th apprx. is`, 24885277358456224884575752948576018270432512779451\ 870212212558689832337736603479604/154620921790141879530635928123323615981140888\ 91405907218509745809847935373127758675 -51 the error is, -0.3539475521 10 `The `, 64, `-th apprx. is`, 12797277030058280184981857907381084311761727600108\ 451528155329588609868626159156636348/795139528601248490580177553991934718862176\ 6936539843103481060271538922739029821482225 -52 the error is, -0.5164461759 10 `The `, 65, `-th apprx. is`, 14183082163044796173525639927170395396568226289398\ 51591390967219688807594904618356429332/8812444427629035960191486557424011739050\ 47637719628549712967608330747757293856887101275 -53 the error is, -0.7535465401 10 `The `, 66, `-th apprx. is`, 22107465419523476510701292855230018771964106572292\ 166581040246493521869694050312145915828/137361405797185003427934661364752181317\ 75096583141845489374180027432375939413896165535475 -53 the error is, -0.1099496896 10 `The `, 67, `-th apprx. is`, 11099331724718536330216085752870892309251313471435\ 71545171111468393620972878959208802524/6896402550833415063421016308235525999329\ 05255416986533237877033565062288610811421624425 -54 the error is, -0.1604267915 10 `The `, 68, `-th apprx. is`, 26666656257486470504834478143708370312734108361486\ 246891842786453484929938177771039861102516/165689251206690189507727499581835882\ 37560397588252091440442136926251631702407403523607801075 -55 the error is, -0.2340770289 10 `The `, 69, `-th apprx. is`, 59275651202412596791666537115282369526750877056726\ 679341540550719085298175597424238798340716/368300328608294059685492335102684353\ 00557792429997771888129134428415138403324484684777832325 -56 the error is, -0.3415385633 10 `The `, 70, `-th apprx. is`, 12146525534631660184265893015452955619472917702213\ 37458059220003843998521861509171192224870756/7547060648187016443137732940840705\ 46322038829937338739430446655111229141053224726587458381575 -57 the error is, -0.4983331646 10 `The `, 71, `-th apprx. is`, 50937110340943756760889460422637927291655899413129\ 3746024967306197024911830023010999617249064692/31649006120346015154827252770958\ 1785824591226220516515341200667344548371133791285444790632170275 -58 the error is, -0.7271080722 10 `The `, 72, `-th apprx. is`, 24525524921909223647265285290894176958850646660974\ 519023808294091458924924824493022544744861768156/152385654223945960389589808372\ 31420260548909345968755868877242674343260711783244945907527490806825 -58 the error is, -0.1060906987 10 `The `, 73, `-th apprx. is`, 26071663903337606467625841674863050503174178263067\ 32813868816240913021990948413835457532727415370948/1619923558524046513937656323\ 145116078058455753828916408792491885169611523167160190643744430006748975 -59 the error is, -0.1547942610 10 `The `, 74, `-th apprx. is`, 15249785655700837789008106518701968920714393634878\ 44077073577548823812659638202470735413372387666348/9475224572432986935061479528\ 48238661727760053446093379458836309988055305865539193664908020319052725 -60 the error is, -0.2258560019 10 `The `, 75, `-th apprx. is`, 10294276526943072062946780317485921725143473254025\ 352670127672153513854614684700044056734007570690558572/639619363220672162915151\ 3979799466076068822548327993455992252011380969778611862715023719008580321627525 -61 the error is, -0.3295396550 10 `The `, 76, `-th apprx. is`, 39176903832419536230373542858365486864684380191047\ 75655007594507065554730175401390065413073566612846292/2434197897896459769557186\ 981758483176836745799500679443603332749365270963983917610825361381624059666475 -62 the error is, -0.4808205538 10 `The `, 77, `-th apprx. is`, 11140072739246322769126434888614892842306872526352\ 66330479669260378352260532131578561997480851562787569588/6921716366428929444517\ 4160053299629078819192001106886308160828673190008444425658495329982163476088983\ 6275 -63 the error is, -0.7015484810 10 `The `, 78, `-th apprx. is`, 17385294083731270201020256479059789260689427285691\ 870675512284932425074884741823012685315544634609200241412/108020905618147751459\ 6110480222577863832425292565371363796216692568144762466700157240759269284177437\ 1074975 -63 the error is, -0.1023603358 10 `The `, 79, `-th apprx. is`, 60297598528450259996642193838075093227277168088989\ 177561965795700752669510027847199208427828617515063230404/374650044357763877110\ 9832900976494439429477159263888215525642361967177322601421065532206467505047880\ 6334575 -64 the error is, -0.1493499525 10 `The `, 80, `-th apprx. is`, 24784027737483968256606259950088932743824145294489\ 993531179334509110091763196816844188623873994850881131306588/153991822523932053\ 8156740316473632685294139281159694105842964480432366408352195689159591318183768\ 6448599990025 -65 the error is, -0.2179103576 10 `The `, 81, `-th apprx. is`, 18757658147482946486990927762215844166324309238724\ 0019740284153399489529625391753767216989010580658622929989764/11654788297557885\ 8621596229530535560672615812601692796922674894771289120382286448772578010276156\ 659692015616575 -66 the error is, -0.3179435929 10 `The `, 82, `-th apprx. is`, 86970015535985102938000394106207246490843092594561\ 7458000491514324844201072138685522978108378361402752234745885572/54037508911699\ 7264302915663292285440366547881157417491831011934521721936483455483043735893106\ 211378326847021934975 -67 the error is, -0.4638971701 10 `The `, 83, `-th apprx. is`, 42497769165749167998433188741357866456785875862211\ 031616115442570121444973537069068613423307166868097096171942404/264053486235302\ 4791272287569267542110867153219293141762493293904259328964983049925923782679610\ 3899559750184068575 -68 the error is, -0.6768506189 10 `The `, 84, `-th apprx. is`, 39102289900747012526415206696083471128912881341118\ 95065996809253983656754881557899546182647294933657611868895446430508/2429561873\ 6611614832952237235374937757123275867224581426436605989095739038967318925210495\ 22369539985899849010880861525 -69 the error is, -0.9875598350 10 `The `, 85, `-th apprx. is`, 82231223075923064932500920699661918429861257594991\ 5275222649449924169919031997486985754669725745043858654798618134324/51093131608\ 6354981810981184541210134711326785517969439527176079612701432885732251345308346\ 056246759957017292483917075 -69 the error is, -0.1440898887 10 `The `, 86, `-th apprx. is`, 14125086288563790391380385626206936548146811675454\ 0040943336204716430690460440141361905547684627177061047484359058876556/87764095\ 6475365277195678349460492888510127728194057111155861534081013948784170636401049\ 85315829536790090870160868124925 -70 the error is, -0.2102340759 10 `The `, 87, `-th apprx. is`, 49019650721871067304150988341038385460591248676682\ 9392371576327064959362129199850501375027658068856125761648338773151612/30457621\ 4734087874606349538269541918365181906378631881324037062675205468958527424707593\ 006389685765322471956363223244225 -71 the error is, -0.3067412960 10 `The `, 88, `-th apprx. is`, 22967426485317711725273580529712549592074333709269\ 993646673414910180600406840736205987927629537576247234835076704734137564/142704\ 6443225757550658701933251016745396867036316478310255180803588003984455435380983\ 8968312118966044672631634029700734825 -72 the error is, -0.4475493553 10 `The `, 89, `-th apprx. is`, 77772383179473738564690471716613202536683719799061\ 61868141733755711280481450652310697232386461312544281155160881705773372/4832269\ 8613363369679904644461992850147921671913404553411181190972197172133133243577097\ 39148447865008472394264993331622225 -73 the error is, -0.6529939965 10 `The `, 90, `-th apprx. is`, 19272977968584441411704020658741098698980915452100\ 297738474292191291403971274910487500275419659560499670512671594187732385444/119\ 7497450487925407668748512757675089655413217501509262946355744207274774568991710\ 6373463401549930387676493736873123085787575 -74 the error is, -0.9527457511 10 `The `, 91, `-th apprx. is`, 32258968331312609815039588231061493238340578758924\ 62410969501540488819264290390183868681963871426102230157546896848651272612/2004\ 3623977096710232227426682592583013444065281914496104054982569058845628396922711\ 39196759460738646820317343542168658761975 -74 the error is, -0.1390094888 10 `The `, 92, `-th apprx. is`, 24280916625360470246011178311531667588984632407498\ 7991345735421447759246267625956051618269672920803265585783027672244333001284/15\ 0865817424657382248628332781976385927584838384053361214152404908161338311509554\ 583034623339370550087113876137416252760922575 -75 the error is, -0.2028203270 10 `The `, 93, `-th apprx. is`, 18545254346249477356392636773729071122211043731357\ 975728110512158923002673480036613248389756720957591713042688694357550520884956/ 1152281439561827552549894271017655885750834734457521609739764863947393256672893\ 0003514404908636362928300116107521886914908744425 -76 the error is, -0.2959225977 10 `The `, 94, `-th apprx. is`, 86923491827506491794063258825986115491848374623512\ 3355852704993565312314057382628408734384593434217539543583157807623915397311652 /540086021063366981180191924130264801832643375019456218625404169175843526892377\ 604775580243605434075116424046342157991890909713975 -77 the error is, -0.4317619976 10 `The `, 95, `-th apprx. is`, 13122148553841139956074385880213912358683527820700\ 5149520707217601623803649516032055548683580454773364979550795492250366140600988 /815324931298238945367613966214639021810592002702057092675807620298148295877938\ 26280399102383143377445687666133199924779428268025 -78 the error is, -0.6299561810 10 `The `, 96, `-th apprx. is`, 42871899714606893461470107957365295153962266144725\ 8344602621465939446240192235788090513195263528188207498391505968610324415306611\ 6/26637809003621514412576189407421752287573505015852888873390030791121638715217\ 12829451500229037733905569461894047729997362304330175 -79 the error is, -0.9191278452 10 `The `, 97, `-th apprx. is`, 87339003849257696489932981732352806352205908727529\ 6979254455101949952606949122643068465193110394723892179992398644248301766304363\ 844/542667742411802148870106976447523961172970665941010450366732178786008298163\ 311179220768202931067312969569915265568153542888703906575 -79 the error is, -0.1341038266 10 `The `, 98, `-th apprx. is`, 15950781512138233359640194870822447861350894557295\ 3510026606190890769973528868948714200290984918117460475442351748896654505965076\ 27404/9910777787019075147591235410786381607838058584820582845530935805046009138\ 727581024851898483996864864804882743591293493287249334763325 -80 the error is, -0.1956618221 10 `The `, 99, `-th apprx. is`, 45978059445969526586699056526423064579030206653292\ 2814013962138244332216002729007549813091737116647513006973984469825114760464989\ 6161244/28567774557039417467120163052652770310235317186115360009166502800791724\ 40829214680532145813836236907060087630083472697185999412712582825 -81 the error is, -0.2854767224 10 `The `, 100, `-th apprx. is`, 6467236986556852675130214510455137293298391117269\ 4531018385608233705380163466629825319472572745627149698253184303177990677289348\ 3778495684/40183202698238031444694073078042225018358941716970985115462108107949\ 2620992265134194030162707364004157722912567911312901417188184335478575 -82 the error is, -0.4165191807 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - ln(5) | <= -------------- | bn | 0.5965671526 bn since , 0.5965671526, is smaller than 1, this sequence does not provide an \ Apery-style proof for the irrationality of, ln(5) but this sequence of rational approximations is still neat! since it is effe\ ctive! This ends this article, that took, 0.446, seconds. to generate. ---------------------------------- if the input is: DelSeqIntV(ArcT(5,n,x),x,n,0,1,arctan(1/5),A, 100); the output is: Better and Better Rational Approximations to, arctan(1/5) By Shalosh B. Ekhad Consider the following sequence of definite integrals for, n, =0,1,2,3,.. 1 / (3 n) (2 n) (2 n) (2 n) | 2 5 x (1 - x) A(n) = | --------------------------------- dx | 2 (2 n + 1) / (x + 25) 0 It is an elementary calculus integral that its values, for, n, from 0 to , 1, are: A(0) = 1/5 arctan(1/5), A(1) = -3 + 76/5 arctan(1/5) Note that these are linear combinations of, arctan(1/5), and 1 with rational coefficients. By the amazing Almkvist-Zeilberger algorithm, A(n), satisfies the following (\ homog.) linear recurrence equation with polynomial coefficients -4 (2 n + 1) (4 n + 7) (n + 1) A(n) 2 + 4 (4 n + 5) (102 n + 255 n + 134) A(n + 1) - (4 n + 3) (2 n + 3) (n + 2) A(n + 2) = 0 and in Maple input form : -4*(2*n+1)*(4*n+7)*(n+1)*A(n)+4*(4*n+5)*(102*n^2+255*n+134)*A(n+1)-(4*n+3)*(2*n +3)*(n+2)*A(n+2) = 0 it follows by induction that, A(n), are all linear combinations of , arctan(1/5), and 1 with rational coefficients Since , A(n), is obviously (why?) very small , pretending that it is zero and\ "solving" for , arctan(1/5), yields the following sequence of terrific approximations to, arctan(1/5) `The `, 1, `-th apprx. is`, 0 the error is, -0.1973955598 `The `, 2, `-th apprx. is`, 15/76 the error is, -0.00002713879725 `The `, 3, `-th apprx. is`, 3350/16971 -8 the error is, -0.2723017285 10 `The `, 4, `-th apprx. is`, 46991/238055 -12 the error is, -0.2661735052 10 `The `, 5, `-th apprx. is`, 233667674/1183753445 -16 the error is, -0.2581598315 10 `The `, 6, `-th apprx. is`, 96227477390/487485521271 -20 the error is, -0.2495471244 10 `The `, 7, `-th apprx. is`, 395041448321650/2001268157308497 -24 the error is, -0.2407995871 10 `The `, 8, `-th apprx. is`, 242828912084544790/1230164003026289331 -28 the error is, -0.2321199517 10 `The `, 9, `-th apprx. is`, 33735304451494701698/170902042969712119065 -32 the error is, -0.2236067578 10 `The `, 10, `-th apprx. is`, 101190966035821878664534/512630406240026708228595 -36 the error is, -0.2153105026 10 `The `, 11, `-th apprx. is`, 744633673047584296861171550/ 3772291907750498021634703959 -40 the error is, -0.2072571613 10 `The `, 12, `-th apprx. is`, 710323678050283466352827990/ 3598478499670835198851964531 -44 the error is, -0.1994592364 10 `The `, 13, `-th apprx. is`, 651205688158529964296012954396050/ 3298988531726709662693171156270433 -48 the error is, -0.1919214185 10 `The `, 14, `-th apprx. is`, 159592706670791997547935091848618734/ 808491876879916574461569773042097375 -52 the error is, -0.1846436894 10 `The `, 15, `-th apprx. is`, 1748764055459544366732003625207323082/ 8859186380835914990597937815455972125 -56 the error is, -0.1776231176 10 `The `, 16, `-th apprx. is`, 161450866988057805301696006588903171999858/ 817905261449857954635326416639387728893265 -60 the error is, -0.1708549436 10 `The `, 17, `-th apprx. is`, 15820181611934820031283140835016854899891827742/ 80144566696262375934327068226899196235023359415 -64 the error is, -0.1643332611 10 `The `, 18, `-th apprx. is`, 782847555341821240824828751108189409138788316858/ 3965882292069672098521247016681578010377290225365 -68 the error is, -0.1580514562 10 `The `, 19, `-th apprx. is`, 1552187897705740489499495572637507510447363489644886/ 7863337447337613590570157033461706222956747907109875 -72 the error is, -0.1520024976 10 `The `, 20, `-th apprx. is`, 259192334500945169785537080081599872287662056370203894/ 1313060611383867159831974023200627364269298832681623875 -76 the error is, -0.1461791337 10 `The `, 21, `-th apprx. is`, 23871202247570725061255638962864230989931056582453962286/ 120930796344784880157591705877153919961399210689453366655 -80 the error is, -0.1405740255 10 `The `, 22, `-th apprx. is`, 4265320810170082099890522912687519137489175400013892938942/ 21607987603236145797999200832962854103900999453069444763015 -84 the error is, -0.1351798405 10 `The `, 23, `-th apprx. is`, 2266514424688537161894895936399581121229524762701927738167315902/ 11482094259932799119289270874648711974545538484358539580935612935 -88 the error is, -0.1299893165 10 `The `, 24, `-th apprx. is`, 565274045999374897630211617841368724388405847487594421944472493994/ 2863661403677294341550568953920052144719848806239389983002344106125 -92 the error is, -0.1249953071 10 `The `, 25, `-th apprx. is`, 42446682358409124026251935017089430195791127163789163632804121693870642/ 215033622796226056986095176242881475313191312842357890950396676210424625 -96 the error is, -0.1201908122 10 `The `, 26, `-th apprx. is`, 670530149018177512289823121074613441485251781672311658037773870856308770/ 3396885672241642179339708251116445760286627046555497923989618146027285353 -100 the error is, -0.1155689999 10 `The `, 27, `-th apprx. is`, 8316188635492466129281374765744659149430237282057971567435933558823560603250/ 42129562801802249996348904435970927422516198197138134560864455933008099937361 -104 the error is, -0.1111232200 10 `The `, 28, `-th apprx. is`, 386988441485635344150926347486803835796880467555075160646651510649835647954710/ 1960471865628284110624869467113213610566645871114025994612795266732252199655379 -108 the error is, -0.1068470130 10 `The `, 29, `-th apprx. is`, 73738229180393805269898538833459208529544515875129\ 56904715806200726310187752618/3735556627336080814889719235321299006395496413113\ 0847549999425728356887771439885 -112 the error is, -0.1027341155 10 `The `, 30, `-th apprx. is`, 29524463452271169856391730924398185932772739642231\ 1043514173283243003980556621104354/14957004845866093493471087951272773225940146\ 41882172871558450116540956092341513276105 -117 the error is, -0.9877846181 10 `The `, 31, `-th apprx. is`, 63522844953723464957539812542801710483959132014780\ 4490287612633472935390840428029699250/32180483189202716755141075259662817305161\ 54734136559339497628121190417349576354283728497 -121 the error is, -0.9497418446 10 `The `, 32, `-th apprx. is`, 73993332369393661890096773896847890864604878234079\ 362100536664911172738717482308805088530/374848007856233243321035873696801184150\ 651136031428153486887121224319004885663948848578337 -125 the error is, -0.9131561260 10 `The `, 33, `-th apprx. is`, 15265817832170810181117773398766571388967838871243\ 836447603157289079351870623714949458186215250/773361763748914024665830612440092\ 32018690313994942393715647405499937012212604991721580201940801 -129 the error is, -0.8779726940 10 `The `, 34, `-th apprx. is`, 12104877133588981093462899125777104122164723961994\ 2269876031456709631943387576468039023286532342/61322945373212726504104039609179\ 6146542578482834487483108575212548208231288539802569199330350915 -133 the error is, -0.8441386866 10 `The `, 35, `-th apprx. is`, 61936888318065260025055750109222189070119457898576\ 065555445308948330436105388004398306675565091126706/313770423028604280542168717\ 753139285127762353736820389294978426085753875760610626321791243700385826545 -137 the error is, -0.8116031069 10 `The `, 36, `-th apprx. is`, 33475228945160238926192301983510286403580959773664\ 518499325347618288372469748672004405433532205945510/169584508236244709327835981\ 826637935538976100386089644806046031868244275303570495103084739550088811099 -141 the error is, -0.7803167791 10 `The `, 37, `-th apprx. is`, 53350593542508066004474164737416879471228359918790\ 9427007222696728508223555363123210471267086816566505608750/27027251060297997751\ 1227723740798532751584229638295378561681783268835541864092533447463848720705472\ 3031274879 -145 the error is, -0.7502322996 10 `The `, 38, `-th apprx. is`, 19591230283030301917805210664409231305903432085425\ 48828877392372057462663621717870755890422753482749480305801530/9924858643188035\ 4582462027129624811802785283761896838620061080748581538532616491612890698752632\ 47643494599517717 -149 the error is, -0.7213039882 10 `The `, 39, `-th apprx. is`, 39434730434424231733096512109318990400232310454071\ 26310100845644992921450946715380161262118647118725567997180338314/1997751644688\ 1747451489775816848788484114375549299802864058984773346748203666315103538275665\ 885297018100254238816125 -153 the error is, -0.6934878363 10 `The `, 40, `-th apprx. is`, 10447968933603663378609286399356995254307304296895\ 344794543740930908598464529388934017918378289019563437058643458642/529290980078\ 2418526384870495904847299694866831626693998693300147074045946695253369215149166\ 1310386368253172601874625 -157 the error is, -0.6667414556 10 `The `, 41, `-th apprx. is`, 50540672349448129897447483496911793409465743367568\ 53992749058698543895840325378310583692457024858169669342112461166581338/2560375\ 3391355048876151189567159951781701778553385151520112684703880839025383646487732\ 893160958659855053818452633645526565 -161 the error is, -0.6410240267 10 `The `, 42, `-th apprx. is`, 20829277675748095740469292579590964932716762613175\ 5519139013663449092080005676946201210298396302072161336945027169270337242/10552\ 0497480230825853512707306878724471997390728250320725046686276938509062742162211\ 9518046100245545908937382084195637182565 -165 the error is, -0.6162962475 10 `The `, 43, `-th apprx. is`, 52612289935579045705184778004170468042543733967638\ 324663999511070197555110362377936348571345103519240929587405677027023230726/266\ 5322866207865589408079269629988857423996344336064188633666988146159447158143664\ 30912672017593007233299205013012481336818955 -169 the error is, -0.5925202825 10 `The `, 44, `-th apprx. is`, 19317500399190965709764496345087170094919611602735\ 7957842771858629274871182970412036505114747351275633803043682980146927445180325\ 78/9786187903052083225331012101486999766342823256440614188374943914817270497623\ 6680464036340090948604831661195976174598142566256269625 -173 the error is, -0.5696597130 10 `The `, 45, `-th apprx. is`, 15581951817072659382710827818965320098686897382386\ 9357421820067140943200670160736803380387836224764497804995769605857468600161718\ 59726/7893770168347620025215116770038643170028773472867834371151507446543222744\ 2098631624221905388514366491040932733031617370518331270027375 -177 the error is, -0.5476794877 10 `The `, 46, `-th apprx. is`, 82272638257200612002633122507463673267051080502761\ 6203655724115660933415224114523290573177195378830196585876782260349412033827868\ 923602/416790723761815713444390423421875166097973761486403276443080721603713145\ 0679000051438491596893895893171857857793083294192613709648944385 -181 the error is, -0.5265458753 10 `The `, 47, `-th apprx. is`, 29703653468152637221714221937103751163946013712317\ 0854393812409483230707933120083496343580631053269396512568493151682492755906386\ 200880678/150477819717638296422951231389158220258794257526771846611497362252216\ 7589530657176018148033320424759797539934002816788120834945436528465435 -185 the error is, -0.5062264174 10 `The `, 48, `-th apprx. is`, 28473095547828248751038972289045059728960254329926\ 9846848793650740798144091239085056258592715304758191236517501681420136050315415\ 591843404878/144243850112343084817551763377720120530999124638371059498809425482\ 4735814936085015991669113112443005705799041027079695650363845253807075622415 -189 the error is, -0.4866898828 10 `The `, 49, `-th apprx. is`, 14234792681374660838599219198377944098267988990897\ 9944868578495880265033656878616749805997046279102343353854405286266999414204629\ 40573359038118/7211303380988009449540496182430043911294832955718412262307288443\ 7179135165128244161202404361288879952487553854023177967886661118673342351222875 -193 the error is, -0.4679062238 10 `The `, 50, `-th apprx. is`, 11847714344485263809237524518183106155635116477448\ 8491553497966736168768306908000157617703977595035765403185121441242270903464731\ 0160043986187023794/60020166378085939057693205587903069994975349489822997660359\ 6156932305740875835687079946925298981291911639477506897025349023090658322205586\ 4502608625 -197 the error is, -0.4498465328 10 `The `, 51, `-th apprx. is`, 20361485541110910197416796739447023662468782684309\ 6766547571111597845229711832193210150035563742874979262041398115386243842197630\ 4308666558130396850/10315067652279418812983925452305073692290453682411549497344\ 2772903905539697369834581327905866298398008009169423254152545119164425545165526\ 59783074673 -201 the error is, -0.4324830005 10 `The `, 52, `-th apprx. is`, 48805401088075419878218339984834529607056104612653\ 5257592869902513525051626546174226542116402087066243179175926287318602805188212\ 767437997443768359896410/247246701623845577993226511260633939871129307519609985\ 1186975618870681422793984553405894272242930023018784714416223109064472061370799\ 929064193325925767589 -205 the error is, -0.4157888759 10 `The `, 53, `-th apprx. is`, 11912227897017378953302578470595782303930429064591\ 0298122593549231570431001864434509994496409821048799574044040975068310155313559\ 374810795553436023642250350/603469901049275036505603584246240402101215364383799\ 5627681557623916892761319415536667302197903050488078435237900775373027341442543\ 86326404916032169546578847 -209 the error is, -0.3997384271 10 `The `, 54, `-th apprx. is`, 20518687551705769118277007825106676357600160842222\ 8374638013621184096721067256603757262569701000180011752064152967102435970259209\ 9862504425763810163277951397746/10394705720488456029881995625084908453762462419\ 2127824061445363010182350654854262466874678207539469106643859835514339816605746\ 88148509382792538903574105660906705 -213 the error is, -0.3843069036 10 `The `, 55, `-th apprx. is`, 88736540814425745624507065503901539371329934649473\ 6056489221558121723378319100480418217099218080710604040189405347342185573498442\ 35208796057235240085309130251767022/4495366607126819276763627334403969183521378\ 7076995150178321374243586590058812792516787912002385117216318018941923686264173\ 9019372782364728631999560037808792972282335 -217 the error is, -0.3694704993 10 `The `, 56, `-th apprx. is`, 44431358073379287258819713326012447199239837702237\ 4732600708004429722390343958947325821346751191141415223728176148478558591463403\ 03997536780728210332729745878088581470/2250879305855173069666432809551254340468\ 1702246432744263254763014000692900715190008277704329319530992496032142344686442\ 8880904666759130610878210829594162806503098063703 -221 the error is, -0.3552063180 10 `The `, 57, `-th apprx. is`, 23179422738990367538233486594646794520357773171929\ 0279934080035958045356185533370759296373401878940346491093259300947963956459726\ 2883783086398855245664214304369358443950/11742626205279545773491026845039315061\ 8784313279820474585851875235656577451564561812824605357065922684670993717331726\ 03180811815065062727836789819209691980301146971348799 -225 the error is, -0.3414923387 10 `The `, 58, `-th apprx. is`, 77981222115063343558335404889159347802896095544518\ 6826038793394667327163034675249970723954181167504945936311165225090887728948563\ 8971936593843871094125390905444116543453059970/39505053798762257229362784047206\ 3528873047266278505692530865400152491426857749100115309612133637079995929643884\ 77722900823720155080319192690236391029198028967682076911563321753 -229 the error is, -0.3283073831 10 `The `, 59, `-th apprx. is`, 87364108872160686126275873689845107556329527115681\ 7082945866931847044144531180238910251878107044515331611619563741059609221000567\ 0190506391022157200307487345854539512347289886/44258396155719538368958801951778\ 9451790524529270765166525068775389283032089567923633669257807694224336351394729\ 11534214398369668816965955216546558039306324816945697453171635255 -233 the error is, -0.3156310837 10 `The `, 60, `-th apprx. is`, 15946477286018151336890350352862221886560443828117\ 1209905422257098468890325684013871737277648881965950016413794534893180019099161\ 243931921327767448590290409364009653387953825909042/807843768023224064310419858\ 4202856261809667613621681309393782529658002728239882992027558848279407341660397\ 22961499130051524835783203869239573237024507832733400088214733312086353585 -237 the error is, -0.3034438534 10 `The `, 61, `-th apprx. is`, 12902425694458859391372789616131554638960042147709\ 3099521732707429146471814471211656318106476893474082067079104331601139220449504\ 742011866429609423109060695432841143567599136933296950/653633025194241896627146\ 2486336340892012777711323973132730784682945566484123241100036162743007621880663\ 6017377474154611865213243377530337022553105699110934728648370485697481764580029\ 1 -241 the error is, -0.2917268556 10 `The `, 62, `-th apprx. is`, 78959326094627423560552290766271237528786742308403\ 7062759845452770364168046469769718575018865560743604625254944897481417696258887\ 154342076527457438179842723323524291259145352577419372210/400005583482606996837\ 7556266772550681099135212449400839365556686200479428799365990873177320403693681\ 5688351678294933665037272856259899586672079357974078657520913013881570022689701\ 23937649 -245 the error is, -0.2804619765 10 `The `, 63, `-th apprx. is`, 35462834583431073224717592593826135038891600993912\ 8612381407413559743456734825269910454542394260279901616052504839961496058780899\ 956744338055970684209920906578452598581494490112764351450/179653658929312008128\ 4079972841523542705726911026317237618294390654804004328301675416655033207901399\ 1914764869101853912772463612753999471045899979065791183316894395833818502191019\ 52614117 -249 the error is, -0.2696317971 10 `The `, 64, `-th apprx. is`, 33397499944992659429688202234971833396693007280825\ 7106432555721670988549699545454958824699155275241321418977854789001737770956051\ 4169907182594332776947378200574138533076075322317053814303646/16919073544709640\ 0374050221974196700737723361668246413972427525012437045494913952419852808444469\ 1223763012027563386876587615334498936356238575822291622453213872902300627641158\ 9151401790784375 -253 the error is, -0.2592195669 10 `The `, 65, `-th apprx. is`, 66475037785575813743412606004095781245704376610841\ 3979311505444577149840296311873866223672257834790730939460931359230372032577986\ 326959596878532852965072924494985885193639026165732104187383407000994/336760552\ 4467219652133897512532849972623253167050703773770126606470735822636165013240067\ 5388104205268332386006136225502021663159658740836366575599487648803400461279720\ 96505159971252369375046619715625 -257 the error is, -0.2492091789 10 `The `, 66, `-th apprx. is`, 20828330214171470911567534103741474233224584964695\ 5310286290540743156193756593849301746015372556012851632782585343702157336482349\ 56658216564052720447295735647069065452861184188568416584664922909466/1055156976\ 6823229167655885504038468498395462170836278161429853984451330399091425736741960\ 8086130556803181013837774967568218730104078653547973569239393981730592556130509\ 352346895549801937766092405125 -261 the error is, -0.2395851447 10 `The `, 67, `-th apprx. is`, 11510022007805890744722593483743541078372403369003\ 5857487244744804779050594026806420497515364370727727443706442631244612211286041\ 19674092379769504573187950763255321630185962192764902812511176569650136834/5830\ 9427104435670799876863559359839074878544025912068381140692485839476155212764704\ 2549850872064867344780895986663906028925884869327076887610900986766056057141361\ 60611055165778941897604348000750695137625 -265 the error is, -0.2303325710 10 `The `, 68, `-th apprx. is`, 38424589958722291121046433538527259299553042033675\ 3980691834969237604494005264915598747429577731712564762021599923245002018741547\ 781529201335998589677045858212197904320598753617432487325801064623647607518/194\ 6578230429507932883153925037365229685879674726983911326078608021206790830734189\ 6652986918538535908509077610954405747474776912399533390312258409234941056154728\ 23516300759116902639311047669109751443620375 -269 the error is, -0.2214371370 10 `The `, 69, `-th apprx. is`, 73133192618583487930279821944230709782742029935443\ 1206856504864086830497196254882225929443817944909967519482691701653109678584258\ 2803254717601107336810808376962104434455395937662490256946001039211720926974987\ 4/37049056561455207336673477638009325591070938417974898108232457100807736224562\ 0871999559649265251780326840401014367084228370042159802189596920415304818842334\ 278764154245886621919603380041091097266939134092940625 -273 the error is, -0.2128850727 10 `The `, 70, `-th apprx. is`, 50722346466877635376853991563191738334036436921867\ 1341616760382396893571615575767368078466808736513997220510344223745440358823503\ 6081790835173195103219431687715385297749843791166260157721947682995138196369548\ 48806/2569578895566443282104721478150873457988128210367768098828139775288416721\ 8289570461361685872928492323702426403846730193445769119831616402637172915296424\ 19821594956531419597081697012559696831954714666268808895246875 -277 the error is, -0.2046631376 10 `The `, 71, `-th apprx. is`, 10776256785900171833086327236794747617590175478851\ 9608846188194505503186431152863425174076796598990912638315586940474053990930911\ 0560557190287620119144461570027694524989511343055574370221692088892727060832082\ 46610074/5459219444497895830119659442738828340831705107322055453161642351312411\ 9795287274455140165876351312867905307402604138819085764886777268929954960110804\ 9271871630814499219826104702518074438725340912892273201100549741125 -281 the error is, -0.1967586004 10 `The `, 72, `-th apprx. is`, 15221029972121496370830948687294831884313574440865\ 8758673249899087099914060366460694058634871400397736563917434148887421101008585\ 6533417318492313923246822708792260346935368720577502759201673425371535303143581\ 5788208778/77109282415962564556897357400473496144799244156362876585955931840761\ 8090829205900510895114467633133546540411873697243609595612267258831399926864296\ 01261664634768926239163054025979868160408226351776218716493532201472125 -285 the error is, -0.1891592199 10 `The `, 73, `-th apprx. is`, 46864543084139167281869032610305727535087415562740\ 1416136064121324219743546866101024891798476867509547557551152215931810569901348\ 9330391376088664541807783001219848990482077957763986818006222825197542284353361\ 14997803346446/2374143730475986031085229669253139573162296679424832770873983545\ 6201918407525842463801773032176753436032070060635678398516503057546993390511645\ 3377199331195173272529396088121277372048797101453099826085749379786011973593137\ 5 -289 the error is, -0.1818532260 10 `The `, 74, `-th apprx. is`, 38281699340941742003016466392225169823629806030916\ 6526527176083778017317538591969461155829892117903625783712085805418062625677096\ 1938442325781645396175351301333455292486906853134066302040505686451493479267933\ 586313046491342/193933943448651826413085148717629431714595770139973991557558808\ 5996046323454928612494004345413928091885995088603369397911272887247861258674335\ 2321083989637017737394958111505531569705420144293899809285095410404963642931959\ 375 -293 the error is, -0.1748293015 10 `The `, 75, `-th apprx. is`, 43714883351523544547265394183070334090183258530078\ 4957230480578513022348088889036347130540584752988039848524108484192704137183671\ 3466070677001151696492569426233322686678921115518723418036498340008267077261782\ 301544512753796202/221458291081971120612620010602084926783842790590732845128710\ 9794839667318854544178444775920159640035453198945853775531902698702393301131997\ 4714522533815115124200948895330422711390226345319943981369430380891256711989487\ 788678125 -297 the error is, -0.1680765653 10 `The `, 76, `-th apprx. is`, 30887947632798776302962447562984641320125405019581\ 4599875181653987356914034495903758797673570405814088605997050732348352198546701\ 1492856735111536642384639484679757581879928273883424529783847157672476949502438\ 17133574602676263478/1564774185209082094021441876811352150072707598072372902645\ 8147098133092549482666634003048750607747842180027442344675590941614423479039837\ 6087244787598554970578717545851205269483979242705709152484678005814355870536653\ 2654831718115 -301 the error is, -0.1615845550 10 `The `, 77, `-th apprx. is`, 17768098693841058815667795242486907179618221087564\ 9389824943924461373925522025491380018640087812673015023370323580723534018821755\ 6540244938279337155221038138006004760663150026676929384826052457912035915915915\ 704862322658333551611730106/900126563502932412667799658320261715855078269757028\ 5892840346694329552544647129729168210666558936106988469098994484853881045457783\ 7396232637533046693563147710491310087659087634065037054465580477474934597582206\ 88364763383761034129932245 -305 the error is, -0.1553432112 10 `The `, 78, `-th apprx. is`, 47378569773530125486250851275756418906913964372415\ 8030907041337776750687795432912155010812858675210991794320782178266550186922080\ 0920953609709760806620341835913179756101047514440311052295550611038623006946568\ 173693149698041504924918266/240018416876050850408716660522457774629835328800997\ 2270988139773562923181652476900388971396200195455078489361610761469697878780336\ 2321096783746194032713273713543494698564840472167888688011914422112603427030394\ 735325472135908636419410805 -309 the error is, -0.1493428621 10 `The `, 79, `-th apprx. is`, 14034058495760126727091390894961637910980833088368\ 8406753706107371549415977435454553074004423493836106841461061295181008997070564\ 9567070482106545994661456455793412051997787443347500632422253227438238192106183\ 2358141269638499770782000970782/71096120431650146584463809811211379570085973821\ 1262177992709894115759821614469154293722568441403363976347931851985094973689029\ 0397272305745346155505968269184017788263482282902769575348084132558850773402107\ 7344530467593765006337081536168375 -313 the error is, -0.1435742086 10 `The `, 80, `-th apprx. is`, 65675091600810874598800292188997584636523927643942\ 9758123085468336018169667035304795217159455607037570083251056879818692884607193\ 9517432100481676889900618331663446985009792395872773835108833364858580063281111\ 114160512095931600959314013676634/332708049009596541078741792607403041823809979\ 9481340582452667440650818488839955922356341829243799136006564709901541064038703\ 5968485165768352258722670013234734395134728374454597795786046295020634987371670\ 651727113570104483766555476938911481125 -317 the error is, -0.1380283097 10 `The `, 81, `-th apprx. is`, 66440736535004347044926832055468337958753422132117\ 9083540774797008340872385731325715017194559020156538426430696499384365511641619\ 0170647120261564307339606035280772753690047767191872725235762864005366247221246\ 89534709821019883341828519441520162/3365867833376419393741075529761417360226131\ 4748559687070334232972287183021491901456098641580586015942448398009721502573662\ 6659975827567592164957299615445384792262660222743988381840077450519861868192801\ 1348503157344235611084583150193154174466665 -321 the error is, -0.1326965691 10 `The `, 82, `-th apprx. is`, 12234179922020429974051098738159499389663358233426\ 2778708337458223691955539569661181613573887073363392983007651893199957823714399\ 5268491341321034084308419858223991196656260891713032759525486854421101797332572\ 746613323184223919796651942526481202906/619779894305855650710668210912968041825\ 4023351176444298339796056558827427807373118618522597638893846434692104429013687\ 8443541343844491870386267654084124463771235716872056264587739045232730620812742\ 26135126814462836895939524530702931700351429064805 -325 the error is, -0.1275707219 10 `The `, 83, `-th apprx. is`, 47563270545247030331519748846264498342529301312330\ 0334310836600240396021300430010250257827176257589467627809334705754861092666033\ 2889693172218078143517775881099852448628610803128666026775557130915084884301008\ 397367765271911592570539383713037665068058/240954105459205252188441792101669256\ 2135434919884545369799266532451932040847068947770958589983040927769934383578582\ 6529264210349751120528817124570254655109762531476680681027194764690992344807464\ 424722067701606664223924025579416024321981838470560596485 -329 the error is, -0.1226428215 10 `The `, 84, `-th apprx. is`, 60646091519078632718808099847720474209257136743787\ 1775889637316680428934640992040629477871282291360539589487831693235597183875425\ 3732132978970058916302273765432774301020746011949303580124598905008957388870081\ 998063519067062710392934646413382310427434/307231285066391362794430766357944511\ 5276198434662564694320909581102352541282339136823164004531805992488601623695501\ 5822045698392839001172934982962218051886636169089010002524744713698734890937186\ 586822522308978512663876985105679615277199979618710786125 -333 the error is, -0.1179052278 10 `The `, 85, `-th apprx. is`, 11151527645646115603591144218472707321078878063468\ 3309683099939551082376935832048995437847446145734886398705968982180982044916028\ 0252259807930115929419191735057174086803336353863932869000082789731479466583278\ 172594943417462606850610663328162882391370877506/564933053921113917973843218116\ 5907870739742052078103709422563226552233969758337160677889844658927540094799203\ 7359036671618386494876485040223585540290469469243733685871518721328779878771736\ 63274959697018508726777091523805871738117223931244042041906733957625 -337 the error is, -0.1133505955 10 `The `, 86, `-th apprx. is`, 43236025087096574941654468718450489690808418744847\ 8814429039007365580184527012481349684885447432944228869405090447726485117198391\ 4551069627775743356114123868090972441063970681089889804298390413492164729270961\ 32857234243256059076301809177161238660525673418028298/2190324094421250110158862\ 0614004412347467868784427889792275769135817574800513643801731519814735980314019\ 7694041459502099917432493119508854342474022858465925253568723907381291283239767\ 0369553160174555932923238110852157917854021549815781588057165126642695725688765 -341 the error is, -0.1089718621 10 `The `, 87, `-th apprx. is`, 19289496081632844217239234115713054119004194797018\ 3126708907124345845503110538827131247885294956111970662137820671511591743645253\ 2710960421009197708649825651390015536960574290197558129237393588125658328288702\ 256555466210015244067013337018265372202474425706647748534/977200099956782108465\ 9454169175194536769482071297376193086342028529363507565248626201051418054146725\ 8727735944415106282597369692883219030029283649086205054882215268576748921621542\ 6452314414811264462581085346469661588644852763993195750508528645065101460107159\ 3550755 -345 the error is, -0.1047622378 10 `The `, 88, `-th apprx. is`, 54579181267785552186951290040229271368686245561248\ 4301194095966307423896780153990089600271628885143812470590492868968502529090788\ 5113594007563720258818751125806013707641751321825114214747678338950844885744690\ 46317405927261173891842858371214672835105147541761751556318/2764964992591170590\ 1118566142136102939719520721942302465571317235076385440012519925969861860577518\ 3589242759085947155368757522749452999987152279928303469685878522797602176243142\ 9415138847614780976806698102901481073346998919100328134241374487636166794310356\ 576088667415 -349 the error is, -0.1007151943 10 `The `, 89, `-th apprx. is`, 62393068405276231231425328892600314628985033935989\ 4878339533408988673164271872188091293227878553405125394605256714615879318660629\ 9369001142291996848200904261940350230457215593193707753852011926655339098786106\ 1117538650066087423450534901619820992689075014266298201933995714/31608141770121\ 9490195483256227242336199977269339861819218137245154579525421275146298293581671\ 1371053176384399280342007022364952102647114459698180807086679954358508250955138\ 0706195059162572788099861107503374767774999109071809783297978325433288887015946\ 5299604650514923565625 -354 the error is, -0.9682445539 10 `The `, 90, `-th apprx. is`, 67315764033178836210352413149548852353419774479743\ 4133572194320797763342613477270841769019770981854494914083820583946926308500882\ 5673284982931957369311142307778958764853958399275748590237590510134299734389692\ 7865298288284578298418687454708119815853073875250677534855483834/34101964646201\ 9950793951950345455268099034656010756131735814030265523202830560563062542250708\ 9480381442115295019965917815227729063827677697980524711966798471256022820203976\ 5828382479710003032837784334245235488032057780351935786833483599475743894438319\ 5368356166978572503125 -358 the error is, -0.9308398682 10 `The `, 91, `-th apprx. is`, 24184642974705994328846996526675723660816315047001\ 3503354314384526710210299061354792919218913746972656423852025682743232599187683\ 5763623418311308669630132535870833508749602381165855291857920942203938344339271\ 055307661764918516124888655601516812535578571624565949262797967931298/122518677\ 6901081529351031777620163994567872588504070479435799701539603997739294896748009\ 5200128345003232112733670310930490896871423235577806714511543626170208310610385\ 0172486529650897332166837081549506491509619571784157278688264426603651007474686\ 90624914094989851148522855779625 -362 the error is, -0.8948798709 10 `The `, 92, `-th apprx. is`, 42179771319073787956006518027323998187572235138577\ 2423949012069764921423830553080108869327043413984162846226356966227977249018554\ 7102667320879482574967650245086865519216617271990804851174875721513558976901064\ 56562723510113305521513225606298358680339393364321093645577364284638687042/2136\ 8145945709967640653760940387347581679041444728373213271418412964612061039256953\ 8095871734746460091182915883819531828274091046154216329325866118514324828528156\ 5903953172163818965026024486807069199751397710369445907171366959121472517625341\ 582423871141690228243270409702607366549625 -366 the error is, -0.8603087852 10 `The `, 93, `-th apprx. is`, 34228094195360812962787309191418587690673031558353\ 8325712161922806393268401530682786755395101383239366777136198394831328552586746\ 1492835695425454096726002354838814837838352943960552769825204617169848344078138\ 96975452566707745064379028512679606369647396025619782795244461996200713403478/ 1733985010675582796432669220985077527967588541567581144651612955791596773034509\ 6219830693031117262544691784366331908115120217434003386770728629999721333643590\ 2648567405520417111291278858171506209429143505728727242092233288971556101366528\ 3455937650416088075093829823107702677674609685875 -370 the error is, -0.8270729862 10 `The `, 94, `-th apprx. is`, 86803417440596133585436078580069256293589099142407\ 0677933906638773503757397492765216521441958936594722943144975473567523150591502\ 5272040294534269743838139692119200618770608882328240025958509007939362856415500\ 4838315683102258671226335751197015256478741091986083244616558559496148598300842 /439743515541130188795324361264323270081042385326185866460078115133253150201262\ 1580364277803997803574328662988028440978688250132960424271385078692859365188784\ 1378506313865550666416115356808013583870027058950025294370899411190595989344515\ 5139107306633057889185999233631410072664738701178125 -374 the error is, -0.7951209178 10 `The `, 95, `-th apprx. is`, 38746157174495337527233028907283926581463934832011\ 0409826457039400710030514279041035748953240626465409322264805367037135332445868\ 1651165102105864419748454034504840021613335710325041260295801999162263636343162\ 5565909719347358284249199234961770694149618648026767138051353352045957861210442\ 0702/19628687293656338589573305114821196999891174146084655297637876665319022935\ 6167315875066327146304528053958323038353314578339432002219717608870420863967353\ 6130730977923924953204194179662709163853507111923205131980638452806150516527097\ 248438276407540715845016999513340197752958651555060743084375 -378 the error is, -0.7644030131 10 `The `, 96, `-th apprx. is`, 43676695417709999665147456807527753208679950248739\ 2792624421047036095836676045759510210798773334573625747233362555025361262517336\ 1172914921118263538696047105155394589735669955574232652210138534554723127639073\ 9545004083753797421973127241909483400855665899713900896163527610357569229907733\ 9338/22126483215187873763303350183382140870079855812478072217173430721469350769\ 9714613966832114613141485792631309980918433550357695303956069885951125971959792\ 8210818823571518399991913917050794834336019558647002733872393726252551288745904\ 228736546773129875446364092230154778283821119823171703572125 -382 the error is, -0.7348716180 10 `The `, 97, `-th apprx. is`, 11638158121926967659873823226399550205747164668414\ 3122135110346584888382802126743804105020266142731751993985800453039462292291167\ 8104548934288527485740551201252617635457952020631190653874136343503722220438019\ 9329153720882908866762832905233783762083067580167588034791951761030557843202324\ 61558/5895856082465979527839960033589386898708278585052176631890374557099388032\ 6421980691788540934010117473592959699677243120319822022915425368671769373790482\ 0858513731763047393677188156761574438390382506718872295616897577597487191719476\ 0134531620768438098536280369196772418556848543981746539287875 -386 the error is, -0.7064809171 10 `The `, 98, `-th apprx. is`, 20111806742594903725325695159619211787773136509987\ 1051005042495839323710767548586809028979112212836330970317684398627678875505124\ 5161299518872065823608193867002971587401124904730575358587666842251057807818867\ 7675614004165910309389807474406528187478621572976690288975759342175744007105714\ 005978254/101885811200059030889706165228710855938024684313310320503842562285062\ 9886717170423932430772580711269157729878104378307234766493902418464731646011116\ 3472878682517414533956187730934589657672886203232007062196115918101443375366282\ 8140137698532906175679888437055698736413944537695661639411173734531375 -390 the error is, -0.6791868630 10 `The `, 99, `-th apprx. is`, 10763496466516449661549721372726936511915100488620\ 7970100262147544803318846615777572541959657508615428509067932053432151660760601\ 9059884803099633853074888375792623615656158701304766158858844585420513849437979\ 7468773341885230455850019783756908874017887777013538649689968099690864202878404\ 726089351169934458/545275510487778158380596030149050308399880758928277771765383\ 9299716973084157816595665352374550426052489643502131128927601254268588399524058\ 6634561358714333741897593989267389971805617910734638797626585293569617299857870\ 6067442834023153128192766965551675955028709779074588224853380199768100935137187\ 74803125 -394 the error is, -0.6529471084 10 `The `, 100, `-th apprx. is`, 1898535192781926930928511459952460828227481196329\ 5166134875569437129956680143242363562692404550066557586899575839866801994286628\ 3622143803078814806895092900295693275736152250214374993273465606015730382020185\ 0962478033384756894222515710398325850042417075699926765435265933275940415258884\ 2281194136777538581178/96179224812643311727556693848126694016630904415774512592\ 1483106408209294459948386700902146496839834875522311876871632009828202707734303\ 0422306830824956036198029658701740586359580040823179266922378525750738752831506\ 2620072478810969782229204560033180293722984499475194828933940442016571803815954\ 213638203303125 -398 the error is, -0.6277209401 10 The sequence of approximations, an/bn seems to have the property, roughly, f\ or some CONST | an | CONST | ---- - arctan(1/5) | <= ------------- | bn | 1.292935416 bn since , 1.292935416, is larger than 1, this sequence probably suffices to \ give an Apery-style proof for the irrationality of, arctan(1/5) This ends this article, that took, 0.829, seconds. to generate. ---------------------------------- This took, 3.601, seconds.