Series $ \sum\limits_{n=1}^{\infty}\frac{1}{n}e^{2 \pi i n x} $

Hint We have for $x\neq 2\pi m$; $$\sum_{k=1}^ne^{ikx}=\frac{\sin(nx/2)}{\sin(x/2)}e^{i(n+1)x/2}$$

Thus Dirichlet's criterion works all right. This says the following:

Let $\langle a_n\rangle$ be a sequence of complex numbers with bounded partial sums, and let $b_n$ be a sequence of real numbers decreasing monotonically to zero. Then $$\sum a_nb_n$$ converges.