Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$.
The problem is that I do not know for what function is $f$ converging to. I think that maybe complex Fourier series might help here, but I'm not completely confortable with that tool.
Any help?
HINT: In order to establish whether the series converges or not, you do not need to actually know its limit. You are only asked to establish the existence of the limit, so you can simply apply one of the Convergence Tests, e.g. Dirichlet test.
PS Before applying these tests, it might be convenient to recall the Euler's Formula and split you series into two:
$$ e^{ix} = \cos x + i\sin x \implies f(x) = \sum_{k=1}^{\infty} \frac{e^{ikx} }{k}= \sum_{k=1}^{\infty} \frac{\cos (kx)}{k} +i\sum_{k=1}^{\infty} \frac{\sin (kx)}{k}, $$ and then establish uniform convergence of each of the series individually.