Re: Incongruence? hmm...

• To: mathgroup at smc.vnet.net
• Subject: [mg102760] Re: Incongruence? hmm...
• From: garrido at ruth.upc.edu
• Date: Wed, 26 Aug 2009 07:44:17 -0400 (EDT)

```Hi Filippo,

FourierCosCoefficient[
1/720 (8 \[Pi]^4 - 60 \[Pi]^2 x^2 + 60 \[Pi] Abs[x]^3 - 15 x^4), x, m]

1/m^4

1/720 (8 \[Pi]^4 - 60 \[Pi]^2 x^2 + 60 \[Pi] Abs[x]^3 - 15 x^4) /.
x -> Mod[x, 2 Pi]

1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 - 15 Mod[x, 2 \[Pi]]^4)

Plot[{Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}],
1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 - 15 Mod[x, 2 \[Pi]]^4)}, {x, -10,
10}];

Therefore,

Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}] =
1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 -
15 Mod[x, 2 \[Pi]]^4) for everything  x Real.

Regards,
J.L.Garrido

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