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

Let f(x) be defined by the following Fourier series for [tex] \left|x\right|[/tex]:

[tex] f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{1,3,...}\frac{cos\left(nx\right)}{n^{2}}[/tex]

Show that

[tex] \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8}[/tex]

and

[tex] \sum_{1,2,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}[/tex]

3. The attempt at a solution

I was able to find the first sum by letting x = 0. I don't now how to approach the second part since the sum consists of the odd and even n integers, but the Fourier series is only comprised of the odd integers.

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# Estimating the sum of reciprocal powers using a given fourier series

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