[seqfan] Re: formula for Pi

[Alexander Povolotsky]

>noting that the summand is 22/(n+7/3) - 7/(n+5) - 15/(n+22/3).

Above was my BBP-like starting point
Note also presence of 22/7 and obviously 22-17=15

Cheers,
ARP
========================================================
On Sun, Dec 7, 2008 at 2:27 PM, Robert Israel <israel at math.ubc.ca> wrote:
> This series has no simple relation to the digits of Pi (whether base
> 10 or any other base).  Moreover, it converges quite slowly, since the
> terms are of order 1/n^2.  By the way, it follows quite easily from the
> identity
>
> sum_{n=0}^infty (1/(n+b) - 1/(n+a)) = Psi(a) - Psi(b)
>
> together with
> Psi(5) = 25/12-gamma,
> Psi(22/3) = 699171/138320-gamma-1/6*Pi*3^(1/2)-3/2*ln(3),
> Psi(7/3) = 15/4-gamma-1/6*Pi*3^(1/2)-3/2*ln(3)
>
> noting that the summand is 22/(n+7/3) - 7/(n+5) - 15/(n+22/3).
> A nicer version, in my opinion, is
>
> sum_{n=0}^infty 1/((3*n+1)*(3*n+2)) = Pi*sqrt(3)/9
>
> Cheers,
> Robert Israel
>
> On Sun, 7 Dec 2008, Hector Zenil wrote:
>
>> Is it possible to get subsequences of the digits of Pi from this
>> formula? Of course without doing N[formula,digits] or any other
>> similar trick...
>>
>>
>> On Sun, Dec 7, 2008 at 3:28 PM, Vladimir Bondarenko <vb at cybertester.com> wrote:
>>> 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
>>>> 920249341301972880
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>>>> 288092....
>>>>
>>>> Cheers Alexander R. Povolotsky
>>>>
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>
>>>
>>>
>>>
>>>
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>>>
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>>
>>
>>
>> --
>> Hector Zenil                          http://www.mathrix.org
>>
>>
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>>
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