$$\sum_{n=1}^\infty \frac {(-1)^n|\sin(n)|}{n}$$
I was helping a friend edit a Sequences and Series test for an upcoming state math competition and was recently given this problem. I'm not quite sure how to prove how this series converges, given that most tests that I know (Alternate, Dirichlet's, etc.) are hard to utilize, simply because this series is most likely going to converge conditionally. The tricky thing is that I can't use powerful tests like the ratio or root test because this series does not absolutely converge, yet intuition tells me that it does not diverge either as its plot on Wolfram shows that it tends to $\approx -0.4$ as $x$ approaches infinity (Wolfram Alpha). Any help would be appreciated.
Edit: This problem remains unsolved. Is there any way to prove it's convergence using Dirichlet's test?
