Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Deduce that
$$\sum _ { k = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { \mathrm { i } \theta } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$
One may use the dominated convergence theorem.