Fourier expansion of $x^2$

Question

I found a problem in an older text -

Find the Fourier expansion of $f(x) = x^2$ on $(\pi, 0)$ w.r.t

1) only $\sin-s$ 2) only $\cos-s$

and find $\sum^\infty_{n=1} \frac{1}{n^2}$ and $\sum^\infty_{n=1}\frac{(-1)^n}{n^2}$

I'm not sure how to do any of it. Any help? I'd appreciate it if those who answer could point out the techniques they used to solve it explicitly, so that I can become better familiar with them.

Answer 1

Hint:

By parts $$\int x^2(a\cos(kx)+b\sin(kx))dx=\frac{x^2}k(-a\sin(kx)+b\cos(kx))\\-\frac2k\int x(-a\sin(kx)+b\cos(x))dx,$$

$$\int x(-a\sin(kx)+b\cos(kx))dx=\frac xk(-a\cos(kx)-b\sin(kx))\\-\frac1k\int(-a\cos(kx)-b\sin(kx))dx.$$

Answer 2

Hint: For (1), construct the odd extension on $(-\pi,\pi)$; for (2), construct the even extension on $(-\pi,\pi)$.