This series ${\sum_{n\ge1}\frac{\sin n}{n\sqrt n}}$ converge or diverge? What is the nature of $${\sum_{n\ge1}\frac{\sin n}{n\sqrt n}}$$ I used the limit comparation test with $\frac{1}{\sqrt n}$. Is it good or I should have done something else?

The series converges by the comparison test:$\sum_{n\geq1}|\frac{\sin n}{n\sqrt{n}}|\leq|\sum_{n\geq1}|\frac{1}{n^{3/2}}|<+\infty$ Since $\lim_\limits{n\to\infty}\frac{\sin n}{n^{\epsilon}}=0\forall\epsilon>0$ therefore, the series converges more rapidly than $\sum_{n\geq1}\frac{1}{n^{(3/2)+\epsilon}}$