$ \begin{align*}\displaystyle\pi^2&=\frac{\pi^2}{3}+\sum_{n=1}^{\infty}\left((-1)^n\frac{4}{n^2}\cos \pi x\right) \\ &=\frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{1}{n^2}\\ \frac{2\pi^2}{3}&=4\sum_{n=1}^{\infty}\frac{1}{n^2}\\ \sum_{n=1}^{\infty}\frac{1}{n^2}&=\frac{\pi^2}{6} \end{align*} $