[seqfan] Re: formula for Pi
Alexander Povolotsky
apovolot at gmail.com
Mon Dec 8 18:29:46 CET 2008
You probably have noticed that the polynomial presented by you for summing
could be made BBP like looking
1/((3*n+1)*(3*n+2)) = 1/(3n+1) -1/(3n+2)
You probably have noticed also that fraction's numerators in above are
summed to 0.
You probably have noticed also that such artifact happens quite often
in BBP formulas
Cheers,
Alexander R. Povolotsky
==============================================
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
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>>>> 288092....
>>>>
>>>> Cheers Alexander R. Povolotsky
>>>>
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
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>>
>>
>>
>> --
>> Hector Zenil http://www.mathrix.org
>>
>>
>> _______________________________________________
>>
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>>
>
>
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