TITLE: How to prove that $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?
QUESTION [5 upvotes]: How to prove $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?   I would appreciate any help.

REPLY [5 votes]: Hint
$$
\sum_n \frac{|\sin n\theta_0|}{n} \geq \sum_n \frac{ \sin^2 n\theta_0 }{n}=\sum_n \frac{ 1-\cos 2n\theta_0 }{2n}$$