Does $\sum_n \frac{\sin n}n$ converge absolutely? [closed]

Question

Using Dirichlet's test, one can prove that $\sum_{n\geq 1} \frac{\sin n}n$ converges. Does it converge absolutely?

Answer 1

No, it doesn't. It is easy to see that for any positive integer $m$ we have

$$ \left|\frac{\sin(m)}{m}\right| + \left|\frac{\sin(m+1)}{m+1}\right| > \frac{1}{6m}.$$ Summing this up over all even positive integers shows that the absolute series diverges.

P.S. This is not of research level, but I like the above simple argument.