Given a function $$f(x) = \begin{cases} \pi -x &\mbox{if } 0<x < 2\pi \\ 0 & \mbox {if } x= 0 \end{cases} $$ with $2\pi$-periodic continuation, I have obtained the Fourier Series:
$$f(x)=\sum_{k=-\infty}^{\infty}\frac{-ie^{ikx}}{k} = +\sum_{k=1}^{\infty} \frac{2}{k}\sin(kx)$$
I guess my zeroth question is whether I got this right, but my actual question is: how do I show that (or whether?) the Fourier Series converges towards my original function $f(x)$? I'm guessing I have to show uniform convergence of the series but I don't know how...
Your function has left- and right-hand limits $f_l$, $f_r$ at every point, and corresponding left- and right-hand derivatives $$ f_{l}'(x) = \lim_{y\uparrow x}\frac{f_l(x)-f_l(y)}{x-y}, \\ f_{r}'(x) = \lim_{y\downarrow x}\frac{f_r(y)-f_r(x)}{y-x}. $$ So the Fourier series for $f$ converges everywhere to the mean of the left- and right-hand limits of $f$. Your function $f$ is everywhere defined so that it is the mean of the left- and right-hand limits. So the series converges to $f$ pointwise everywhere.