Find the value of the following summation:
$$ \sum_{n=-2}^2 c_n $$
Such that $ c_n $ are the complex Fourier coefficents for:
$$ f(x)=e^\frac{ix}{2} : x \in [-\pi, \pi]$$
Here is the attempt:
$$ c_n = \frac{1}{2\pi} \int_{-\pi}^\pi e^\frac{ix}{2}e^{-inx} $$
$$ = \frac{1}{2i\pi} \biggl[\frac{e^{ix(\frac{1}{2}-n)}}{\frac{1}{2}-n}\biggr]^\pi_{-\pi} $$
I evaluated $ e^{ix(\frac{1}{2}-n)} $ on the complex unit circle for $\pm\pi$ and determined them to both be the same value, thus:
$$ c_n = \frac{(-1)^n}{\pi(\frac{1}{2}-n)} : n \in \mathbb{Z}$$
With this, I was able to find the summation:
$$ \sum_{n=-2}^2 c_n = \frac{2}{5\pi}-\frac{2}{3\pi}+ \frac{1}{2\pi}+ \frac{1}{2\pi} -\frac{3}{2\pi} $$
Here's the interesting part- using $c_n$ in the actual function it's supposed to represent (I wanted to check my $c_n$ is correct) :
$$ f(x) = \sum_{n=-\infty}^\infty c_n e^{inx} $$
returns "Series does not converge" on Wolfram Alpha. So I assume that I've made a mistake somewhere in calculating $c_n$, is this true?