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Fourier series of the periodic function (period $2 \pi$ ) defined by

$f(x)=\left\{\begin{array}{ll}0 & -\pi x x<0 \\ x & 0<x<\pi\end{array}\right.$ is $\frac{\pi}{4}+\sum_1^{\infty}\left[\frac{1}{\pi} n^2\left(\cos n \pi-1 \cos n x-\frac{1}{n} \cos n \pi \sin n x\right]\right.$

By putting $x=\pi$ in the above, one can deduce that the sum of the series

$1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots$, is

- $\frac{\pi^2}{4}$
- $\frac{\pi^2}{6}$
- $\frac{\pi^2}{8}$
- $\frac{\pi^2}{12}$