For what values of $p$, the series $\sum_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p$ is convergent? Let $p>1$ , $p\in\mathbb{R}$. For what values of $p$, the series $\sum_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p$ is convergent?

When $p=1$, I know the series is divergent but how about other cases? Thanks! 

If $p > 1$ we can write:
$0 < \sum\limits_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p \le \sum\limits_{n=1}^{\infty}|\frac{1}{n}|^p$
and the right part converges.So the middle part also converges as $\sum\limits_{n=1}^{k}|\frac{\sin(n)}{n}|^p$ is an increasing function of $k$ and it's limited by $\sum\limits_{n=1}^{\infty}|\frac{1}{n}|^p$.
So for every $p > 1$ it converges.
