I am studying the convergence of the series: $$\sum\limits_{n=1}^\infty n^{3/2}\sin\left(\frac{3}{n^3}\right) $$
I have tried the following: As, $|\sin x|\le 1$ and term of the series are positves we have, $$\sum\limits_{n=1}^\infty n^{3/2}\sin\left(\frac{3}{n^3}\right) \le \sum\limits_{n=1}^\infty n^{3/2} =\infty$$
but this estimate leads neither to convergence nor to the divergence.
What is an elegant to prove the convergence or the divergence of this series? thanks in advance.