(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the Fourier series technique to show that the following series sum to the quantities shown:

[tex]\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}[/tex]

[tex]1+\frac{1}{3^2}+\frac{1}{5^2}+...=\frac{\pi^2}{8}[/tex]

2. Relevant equations

I know the Riemann-Zeta function is

[tex]\zeta (m)=\sum_{n=1}^\infty \frac{1}{n^m}[/tex]

3. The attempt at a solution

For the first series, I know that it's the evaluation of [itex]\zeta (4)[/itex]. The problem is getting the derivation down. I've found an example of it being solved and proven at [itex]\zeta (2)[/itex] here: http://planetmath.org/?op=getobj&from=objects&name=ValueOfTheRiemannZetaFunctionAtS2 . The problem is, I don't really understand what to define as the initial function [itex]f(x)[/itex].

For the second series, I think I've found it to be:

[tex]\frac{1}{(2n-1)^2}[/tex]

After that, I don't know how to go about solving it using the Fourier method. Any help would be greatly appreciated, thanks!

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# Homework Help: Fourier Series and the Riemann-Zeta Function

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