[seqfan] Re: Conjectured sum
Robert Israel
israel at math.ubc.ca
Fri Jan 2 00:36:38 CET 2009
Just reverse the order of summations, and it becomes
4/3 sum((-1)^(1+i)/i^3, i=1..infinity)
= 4/3 sum(1/i^3, i odd) - 4/3 sum(1/i^3, i even)
Now note that zeta(3) = sum(1/i^3, i odd) + sum(1/i^3, i even)
and sum(1/i^3, i even) = 1/8 zeta(3). ...
Cheers,
Robert Israel
On Thu, 1 Jan 2009, Vladimir Reshetnikov wrote:
> Hi,
>
> Here is a conjecture:
>
> sum(sum(4*(-1)^(i + 1)/(3*i^4), i = j .. inf), j = 1 .. inf) = zeta(3)
>
> zeta(3) is the Apery's constant.
>
> Can anybody prove of disprove it?
>
> Thanks
> Vladimir
>
>
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