formula for Pi

• To: mathgroup at smc.vnet.net
• Subject: [mg21999] formula for Pi
• From: Arnold <arnoldk at gauss.cam.wits.ac.za>
• Date: Thu, 10 Feb 2000 02:25:41 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```The following remarkable identity for Pi can be used to calculate the
digit of Pi without calculating first the earlier digits.
Mathematica 4.0 simplifies the sum in terms of hypergeometric functions

In[2]:=
pi = Sum[(1/16)^k (4/(8k + 1) - 2/(8k + 4) - 1/(8k + 5) - 1/(8k + 6)),
{k, 0,
Infinity}]
Out[2]=
\!\(\(-2\)\ ArcTanh[1\/4] + 4\ Hypergeometric2F1[1, 1\/8, 9\/8, 1\/16] -

1\/5\ Hypergeometric2F1[1, 5\/8, 13\/8, 1\/16] -
1\/6\ Hypergeometric2F1[1, 3\/4, 7\/4, 1\/16]\)

Can one use Mathematica to show that this last expression equals Pi?

(In the December 1999 issue of the American Mathematical Monthly p.903
it is shown how to prove the sum equals Pi using another system.)

Arnold Knopfmacher
Witwatersrand University
South Africa

```

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