[seqfan] Re: formula for Pi

[Vladimir Bondarenko]

Hello,

This is an exact formula for Pi.

FullSimplify[6/7*(1/3*Sum[(843*n + 4607)/((n + 5)*(3*n + 7)*(3*n + 22)),
{n, 0, Infinity}] - 655999/248976 - 7/2*Log[3])*Sqrt[3]]

Pi

Cheers,

Vladimir Bondarenko

Quoting Alexander Povolotsky <apovolot at gmail.com>:

> I've got this very ugly formula by playing Maple syntax via old and
> new inverse symbolic calculators :
>
> Pi = 6/7*(1/3*sum((843*n + 4607)/((n+5)*(3*n+7)*(3*n+22)),n=0...infinity)
>  - 655999/248976  - 7/2*ln(3))*sqrt(3)
>
> What I've got for Pi above - is it just a good approximation or exact ?
> (My old PC with PARI/GP can not get over the summing )
>
> It looks that this 655999/248976 fraction quickly becomes periodic around
>  920249341301972880
>
> gp > \p 1000
> realprecision = 1001 significant digits (1000 digits displayed)
> gp > 1.0*655999/248976
> %4 = 2.6347880
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> Cheers Alexander R. Povolotsky
>
>
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