Computing Pi as Slowly as Possible

Shalosh B. EKHAD and Doron ZEILBERGER

\frac{\binom{2n}{n}}{4^{n}} = \frac{1}{\sqrt{n}} \frac{1}{\sqrt{π}} ·

(1 − 1/8 n^{−1} + \frac{1}{128} n^{−2} + \frac{5}{1024} n^{−3} − \frac{21}{32768} n^{−4} − \frac{399}{262144} n^{−5}

+\frac{869}{4194304} n^{−6} + \frac{39325}{33554432} n^{−7} − \frac{334477}{2147483648} n^{−8} − \frac{28717403}{17179869184} n^{−9} + \frac{59697183}{274877906944} n^{−10}

+\frac{8400372435}{2199023255552} n^{−11} − \frac{34429291905}{70368744177664} n^{−12} − \frac{7199255611995}{562949953421312} n^{−13}

+\frac{14631594576045}{9007199254740992} n^{−14} + \frac{4251206967062925}{72057594037927936} n^{−15} − \frac{68787420596367165}{9223372036854775808} n^{−16}

−\frac{26475975382085110035}{73786976294838206464} n^{−17} + \frac{53392138323683746235}{1180591620717411303424} n^{−18}

+\frac{26275374869163335461975}{9444732965739290427392} n^{−19} − \frac{105772979046693606062363}{302231454903657293676544} n^{−20} + O(n^{−21}))

References

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: ShaloshBEkhad at gmail dot com .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .