Does $\sum\limits_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{n}$ converge

Question

Question：

Does $\sum\limits_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{n}$ converge?

Attempt:

$\sin(n)\sin(n^2)=\frac12[\cos n(n-1)-\cos n(n+1)]$,

so if we can prove that $\sum \cos n(n+1)$ is bounded,then by Dirichlet test,$\sum \frac{\cos n(n+1)}{n}$ converge.

Answer 1

Note that

$$\left|\sum_{k=1}^n [\cos k(k-1) - \cos k(k+1)]\right| = |1 - \cos n (n+1)| \leqslant 2$$

Now apply the Dirichlet test.