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 >