Is the series $$\sum_{n\ge 1} \frac{\sin(n^2)}{n}$$ convergent?
My thoughts so far:
1) This is an alternating series so the integration test does not work here.
2) The Weyl inequality roughly says $$\sum_{n\le N} \sin(n^2)$$ is $O(N^{1/2+\epsilon})$, so the Dirichlet test does not work directly, but one can take $$a_n=n^{-1},b_n=\sum_{k\le n} \sin(k^2)$$ and follow the idea of Dirichlet test. The problem now is that the Weyl bound does not hold for all $N$.