here is an exercice I'm struggling with so if anyone can help it would be highly appreciated .
Let $f\; : \mathbb{R}\rightarrow\mathbb{R}$ be a $2\pi$ periodic function define over $\left[-\pi,\pi \right] $ with :
$f(x)=\begin{cases} x & \text{ if } x\in \left[0,\frac{\pi}{2} \right]\\ \pi - x& \text{ if }x \in\left[\frac{\pi}{2},\pi \right] \\ -f(-x)& \text{ if } x\in \left[-\pi,0 \right] \end{cases}$
I know that the Fourier series of a function $f(x)$ is a series on the form $${1\over 2}a_0 +\sum_{n=1}^\infty \bigl(a_n \cos(nx) + b_n \sin(nx)\bigr)$$
where the coefficients $a_n$ and $b_n$ are given by :
$a_n ={1\over\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)\,{\rm d}x\qquad$
$b_n ={1\over\pi}\int_{-\pi}^\pi f(x)\sin(nx){\rm d}x\qquad$
but I have some trouble computing due to how $f$ is define .. and how can I prove that it will converge uniformly to the original function $f$ .
if anyone could help thanks in advance