Sum of Fourier Series I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by

$$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$

in the interval $]-\pi,\pi[$ and give the sum of the series for $x=p\pi,p\in\mathbb{Z}$.
What I know:
If $f(x)=\sum_{n=-\infty}^{\infty}\alpha_ne^{inx}$ on $[-\pi,\pi]$, then
$\alpha_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$.
Questions:
Can I use the above formula (the intervals are different)?
Should I use integration by parts?
To compute the sum, do I just substitute $x=p\pi$ in?

Here is how you advance
$$ \alpha_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx= \frac{1}{2\pi}\int_{-\pi}^{0}(0)e^{-inx}dx + \frac{1}{2\pi}\int_{0}^{\pi} \cos(x)e^{-inx}dx$$

$$ \implies \alpha_n = {\frac {in \left( {{\rm e}^{i\pi \,n}}+1 \right) }{2\pi({n}^{2}-1)}}.$$

The case $n=1$ can be obtained from the above formula as

$$ \alpha_1=\lim_{n\to 1}\alpha_{n}=\frac{1}{4}. $$  

