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Incongruence? hmm...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102710] Incongruence? hmm...
  • From: Filippo Miatto <miatto at gmail.com>
  • Date: Thu, 20 Aug 2009 04:56:35 -0400 (EDT)
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)!

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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