I suggest this problem to you.
Let $\theta \in \mathbb{R}$. For all nonnegative integer $n$ let $s_{n}=1+\exp(i\theta)+\exp(2i\theta)+\ldots + \exp(in\theta)$. Determine a necessary and sufficient condition over $\theta$ for the sequence ${(s_{n})}_{n \in \mathbb{N}}$ to be bounded.
Terminology : The complexe sequence $(s_{n})_{n \in \mathbb{N}}$ is said to be bounded iff $\exists M \in \mathbb{R}^{+}\, \forall n \in \mathbb{N} \left | z_{n}\right | \leq M.$
Any idea ?