Fourier analysis question

Question

Let $f(t)=\frac 12 -t, t\in(0,1).$ Calculate the Fourier coefficients of the function $f$ and the sum $\sum_{n=1}^{\infty} \frac {1}{n^2}$. Note that $L^2 (\Bbb{T}) \to l^2(\Bbb{Z})$ and $\sum_{n\in\Bbb{Z}}|\hat{f}(n)|^2 =\int_0^1 |f(t)|^2 dt$ .

Answer 1

Hint: You should start finding the coefficients

$$\hat{f}(n) = \int_{0}^{2\pi} f(t) e^{-int} dt .$$