Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2000
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2000

[Date Index] [Thread Index] [Author Index]

Search the Archive

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
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



  • Prev by Date: parametric equations
  • Next by Date: Re: Bracket Trouble with IT Keyb.
  • Previous by thread: Re: parametric equations
  • Next by thread: Re: formula for Pi