[seqfan] Re: formula for Pi

[Alexander Povolotsky]

22-7=15

On Sun, Dec 7, 2008 at 2:47 PM, Alexander Povolotsky <apovolot at gmail.com> wrote:
>>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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>>>>> 92024934130197......
>>>>> 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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>>>
>>
>>
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>