Does $\sum |\sin n| / n$ converge?

Question

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.

Hint: $|\sin n| \ge \frac12 (1-\cos(2n))$ – achille hui Nov 23 '14 at 16:23 — achille hui, Nov 23 '14 at 16:23

Did · Accepted Answer · 2014-11-23T17:03:39.723

11

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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edited Nov 23 '14 at 17:03

answered Nov 23 '14 at 16:25

Did

279,727

Do you mean $k > 0$? – Nov 23 '14 at 16:28
@ChantryCargill Not necessarily (but one uses this for $k\gt0$, yes). – Did Nov 23 '14 at 16:29
Oh never mind, I just understood what you meant. – Nov 23 '14 at 16:30
@ChantryCargill Perfect. – Did Nov 23 '14 at 16:31
My bad .$\ \ \ $ – Martin Argerami Nov 23 '14 at 16:39

Does $\sum |\sin n| / n$ converge?

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