Does the series $\sum_{n=1}^{\infty} \frac{\sin \sqrt{n}}{n^p}$ converge while $0

Question

The original range for $p$ is $p>0.$ It obviously converges when $p>1$. If $\frac{1}{2}<p \le 1$, it's not hard to prove the convergence by comparing it with its integral. But it became much harder when $0<p\le \frac{1}{2}$.