Disprove or prove $\sum_{n\ge1} \frac{\vert{\sin n}\vert}{n} $ is divergent.

Question

I want to prove or disprove $\displaystyle\sum_{n\ge1} \frac{{\sin n}}{n}$ is absolutely convergent.

I can prove $\displaystyle \sum_{n\ge1} \frac{{\sin n}}{n}$ is convergent by dirichlet's test (Can I prove that $\displaystyle \sum_{n\ge1} \frac{{\sin n}}{n}$ converges without dirichlet's test), but I can't prove or disprove that $\displaystyle\sum_{n\ge1} \frac{\vert{\sin n}\vert}{n} $ converges.

Any help would be appreciated.

Answer 1

Hint:

$$\frac{| \sin n|}{n} \geqslant \frac{\sin^2 n}{n} = \frac{1}{2n} - \frac{\cos 2n}{2n}$$

$\sum \frac{\cos 2n}{2n}$ converges, $\sum \frac{1}{2n} $ diverges

Answer 2

Hint: There are infinitely many integers which are more than $1$ away from the nearest multiple of $\pi$ (because in between every two consecutive multiples of $\pi$ there are three integers, and the middle one has this property). If $n$ is such a number, what does this mean about $|\sin n|$?