Re: formula for Pi

• To: mathgroup at smc.vnet.net
• Subject: [mg22036] Re: formula for Pi
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Fri, 11 Feb 2000 02:38:17 -0500 (EST)
• Organization: Universitaet Leipzig
• References: <87tq26\$5ji@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

go to the next book store and buy/order

"Mathematica in Action"
S. Wagon
TELOS/Springer 1999
Second edition

it has a whole chapter about Pi and the ways to
compute it with Mathematica.

But

Sum[(1/16)^k (4/(8k + 1) - 2/(8k + 4) - 1/(8k + 5) - 1/(8k + 6)),
{k, 0, Infinity}] // FullSimplify

will give Pi.

Regards
Jens

Arnold wrote:
>
> 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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