I want to see if this series converges or not:
$$
\sum\_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2).
$$  

I tried comparison tests but nothing. I saw that integral criteria works but I don't know how to show that.


Thank you



How to show this series converges $\sum\limits_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2)$
As indicated by Todd Trimble in comments, we can use the Dirichlet test; here, since $$\sin(n)\sin(n^2)=\frac12\big( \cos n(n-1) - \cos n(n+1) \big)$$
we have a telescopic sum $$\sum\_{n=1}^M \sin(n)\sin(n^2)=\frac12-\frac12 \cos M(M+1)=\sin^2\Big(\frac{M(M+1)}2\Big),$$
that does not exceed $1$ in absolute value.


