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Re: Incongruence? hmm...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102719] Re: Incongruence? hmm...
  • From: Szabolcs Horvát <szhorvat at gmail.com>
  • Date: Fri, 21 Aug 2009 04:43:24 -0400 (EDT)
  • References: <h6j33u$5j4$1@smc.vnet.net>

On 2009.08.20. 10:56, Filippo Miatto wrote:
> Dear all,
> I'm calculating the sum
>
> Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}]
>
> in two different ways that do not coincide in result.
> If i expand the cosine in power series
>
> ((m x)^(2n) (-1)^n)/((2n)!m^4)
>
> and sum first on m i obtain
>
> ((-1)^n x^(2n) Zeta[4-2n])/(2n)!

Hello Filippo,

I believe the result above to be valid only for n=0 and n=1.  For other 
values of n the series will not be covergent.

>
> then I have to sum this result on n from 0 to infinity, but Zeta[4-2n]
> is different from 0 only for n=0,1,2 and the result is
>
> \[Pi]^4/90 - (\[Pi]^2 x^2)/12 - x^4/48
>
> Three terms, one independent on x, with x^2, one with x^4.
>
> however if I perform the sum straightforwardly (specifying that
> 0<x<2pi) the result that Mathematica gives me is
>
> \[Pi]^4/90 - (\[Pi]^2 x^2)/12 + (\[Pi] x^3)/12 - x^4/48
>
> with the extra term (\[Pi] x^3)/12. Any idea on where it comes from??
> Thank you in advance,
> Filippo
>



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