I want to show that $\sum_{n=1}^{\infty} \frac{\sin (n \alpha)} {n}$ converges for any $\alpha \in \mathbb{R}$. Here is what I have:
$\sin \alpha = \operatorname{Im}(\cos \alpha + i \sin \alpha) \\ \sin (x \alpha) = \operatorname{Im}(\cos (n \alpha) + i \sin (n \alpha) = \operatorname{Im}(\cos \alpha + i \sin (\alpha))^{n}\\ \sum \sin(\alpha x) = \operatorname{Im} \sum (\cos \alpha + i \sin \alpha)^n \\ \sum \sin(\alpha x) = \operatorname{Im} \frac{(\cos \alpha + i \sin \alpha)^{N+1} - (\cos \alpha + i sin \alpha)}{1 - (\cos \alpha + i \sin \alpha)} \leq \frac{4}{(1 - \cos \alpha)^2 + \sin^2 \alpha} $
I know that theorem that says that a series of nonnegative terms converges if and only if the partial sums form a bounded sequence. So using this theorem, is it safe to conclude that the series converges $\forall \alpha \in \mathbb{R}$?