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MathGroup Archive 2000

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Re: formula for Pi

  • To: mathgroup at smc.vnet.net
  • Subject: [mg22027] Re: [mg21999] formula for Pi
  • From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
  • Date: Fri, 11 Feb 2000 02:38:10 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

See chapter 20 of Stan Wagon's "Mathematica in Action" (Second Edition). You
can find the formula and the complete proof obtained with Mathematica in
there and lots more related stuff. If you can't get hold of a copy I can
send a Mathematica notebook.

> From: Arnold <arnoldk at gauss.cam.wits.ac.za>
To: mathgroup at smc.vnet.net
> Date: Thu, 10 Feb 2000 02:25:41 -0500 (EST)
> To: mathgroup at smc.vnet.net
> Subject: [mg22027] [mg21999] formula for Pi
> 
> The following remarkable identity for Pi can be used to calculate the
> nth hexadecimal
> 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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