TITLE: Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
QUESTION [16 upvotes]: Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$
  converge?

I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in advance.

REPLY [5 votes]: It is divergent:
Write
$$\sum \frac{\sin^2(n)}{n} = 
\sum \frac{1}{2n} - \sum \frac{\cos(2n)}{2n},$$
clearly at the right handed side, the first is divergent but the second converges.