Consider the function $f(x)$ defined by

$$f(x)=x^{2}, \text { for }-\pi<x<\pi$$

Calculate the Fourier series representation for the $2 \pi$-periodic extension of this function. Hence establish that

$$\frac{\pi^{2}}{6}=\sum_{n=1}^{\infty} \frac{1}{n^{2}}$$

and that

$$\frac{\pi^{2}}{12}=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$$