Convergence of the series $ \sum_{n=1}^{\infty}\frac{\sin n}{n+2\cos n} $

Question

I know that the series $ \sum_{n=1}^{\infty}\frac{\sin n}{n+2\cos n} $ is not absolutely convergent, but I don't know how to determine if its conditional convergent or divergent. I can't use any comparison tests because its not a positive series, I can't use Dirichlet's or Abel's tests since $ n+2\cos n $ is not monotonic series. Any ideas will help. Thanks

Answer 1

\begin{eqnarray} \frac{\sin n}{n+2\cos n} &=& \frac{\sin n}n+\left(\frac{\sin n}{n+2\cos n}-\frac{\sin n}n\right) \\ &=& \frac{\sin n}n-\frac{2\cos n\sin n}{n(n+2\cos n)}\;. \end{eqnarray}

The sum over the first term is known to converge (which you can show e.g. using the Dirichlet test), and the second term can be bounded by the terms of the convergent series $\sum_n\frac1{n^2}$.