I'm tring to show in an excersice the convergence of
$$ \sum_{n\geq 1} \frac{e^{i n x}}{n}, \;-\pi<x<\pi. $$
Indeed we could wirte
$$ \sum_{n\geq 1} \frac{e^{i n x}}{n} = -\log(1-e^{ix}), $$
but I think this is not effective in showing its convergence, since one can take $x=0$.
The hint of the excersice (in fact, the excersice 8 in Chapter 2 of Stein's Fourerier Analysis) suggests to use Dirichlet's test. In this way, it's easy to see that $\frac{1}{n^\alpha}$ goes to $0$ monotonically for some $\alpha > 0$, but I find it difficult to show the boundedness of a summation of $e^{i n x}$.