Find the $n$th Fourier polynomial for $$f(x) = \begin{cases} x, & -\pi < x \le 0\cr -x, & 0 < x \le \pi\end{cases},$$ assuming that $f(x)$ is periodic with period $2\pi$.

$F_n(x) = a_0 + \sum_{k=1}^{n} a_k\,\cos(k x) + b_k\,\sin(k x) =$
$+ \sum_{k=1}^{n}$ $\cos(k x) +$ $\sin(k x)$.