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my job here is to find Fourier series of the function $f(X) = | \sin \frac{x}{2}|$ and the value of $ \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2(2n+1)^2}$.

I found the series: $S_{f} (x) = \frac{2}{ \pi} + \frac{4}{ \pi} \sum_{n=1}^{\infty} \frac{\cos(nx)}{1-4n^2}$, but I don't know how to get to $\sum_{n=1}^{\infty} \frac{1}{(4n^2-1)^2}$, this power of 2 confuses me. Anybody help? Thanks!

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  • $\begingroup$ I'm sorry, didn't notice I made the mistake, edited $\endgroup$ – user389231 Sep 6 '18 at 12:31
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Hint: Use Parseval's theorem, then $$\dfrac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)\ dx=\dfrac12a_0^2+\sum_{n=1}^\infty(a_n^2+b_n^2)$$ then $$\sum_{n=1}^{\infty} \frac{1}{(1-4n^2)^2}=\dfrac{\pi^2-2}{16}$$

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