# Convergence tests $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ [duplicate]

I am having hard time deciding whether $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converges or diverges.

Intuitively I want to say that it converges, because it is like the alternating harmonic series. None of the tests such as the limit comparison test or root tests are applicable on this, so can someone help me out?

## marked as duplicate by Did, Andrew D. Hwang, R_D, Leucippus, Daniel W. FarlowJun 8 '16 at 4:24

• $\displaystyle{\large{\pi - 1 \over 2}}$. – Felix Marin Mar 22 '14 at 1:29
• you can put (-1)^n instead of sin(n) and after your series will be $\sum_{n=1}^\infty \frac{(-1)^n}{n}$ and if you take absoulete value and apply ratio test you will be fail,but we know that 1/n is divergent by p test,however it is condtionally convergent by applying alternating series test. – egemen Jun 7 '16 at 20:16