#C13.txt; March 6, 2014; 
#Implementing Jesus' Guillera PROVED formula for Pi

Help:=proc(): print(` JesusPi(n), JesusPi1(n), JesusPi1e(n) `):  end:

#JesusPi1(n): computes the n-th partial sum of
#https://sites.google.com/site/guilleramath/
#due to Jesus Guillera. It is PROVED, and
#much more importantly using the so-called WZ
#method (that stands for Wilf and Zeilberger)
JesusPi1:=proc(n) local c,s,i,c1:
c:=1:
s:=13:
for i from 1 to n do
c:=c*(-1)/2^10*((2*i-1)/(2*i))^5:
c1:=c*((20*i)*(41*i+9)+13):
s:=s+c1:
od:

s:

end:

#JesusPi(n): computes Pi from 128/Pi^2
JesusPi:=proc(n): evalf(sqrt(128/JesusPi1(n))):end:


#floating point version
JesusPi1e:=proc(n) local c,s,i,c1:
c:=1:
s:=13:
for i from 1 to n do
c:=evalf(c*(-1)/2^10*((2*i-1)/(2*i))^5):
c1:=evalf(c*((20*i)*(41*i+9)+13)):
s:=evalf(s+c1):
od:

s:

end: