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
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Sign up to join this communityI 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
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.