TITLE: Does $\sum |\sin n| / n$ converge?
QUESTION [11 upvotes]: How can I prove if the following series converges? $$\sum_{n\geqslant1} \frac{|\sin n|}{n}$$
I can't use differential or integral calculus. I've tried using Dirichlet and Cauchy tests, but they didn't get me anywhere.

REPLY [11 votes]: Assertion: For every $k$, either $|\sin(3k)|\geqslant\frac12$ or  $|\sin(3k+1)|\geqslant\frac12$ or  $|\sin(3k+2)|\geqslant\frac12$. Hence the series diverges.
Hint: 
The assertion above uses the identity $\sin\left(\frac\pi6\right)=\frac12$ and the inequality $\mathsf{length}\left([-\frac\pi6,\frac\pi6]\right)\lt2$.
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