Does $\sum |\sin n| / n$ converge? [duplicate]

real-analysis sequences-and-series

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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