Convergence of $\sum\limits_{n=1}^\infty \sin \left( \frac{\sin \sqrt{n}}{\sqrt{n^\alpha}} \right)$

I'm trying to figure out for which real values of $\alpha$ does this sum converges:\begin{equation}\sum_{n=1}^\infty \sin \left( \frac{\sin\left(\,{\sqrt{n}}\,\right)}{\sqrt{n^\alpha}} \right)\end{equation}

• I initially thought we could apply Dirihlet test for the inner function and then direct comparison test, but I realized that we cannot easily proof$\ \sum \sin\left(\sqrt{n}\right)$ convergence.
• Besides, the $\sin\left(\sqrt{n}\right)$ function takes negative values at some $n$, the direct comparison test cannot be applied here as well.

So now I'm stuck on that problem.