Is $\sum_{n\ge 1} \sin(n^2)/n$ convergent? 
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 $N$.

You are on the right track. The key is to consider partial sums:
$$ S_N = \sum_{n=1}^{N}\frac{\sin(n^2)}{n} $$
then find a good rational approximation of $\pi$ depending on $N$, apply Weyl bound (or Weyl differencing technique) to estimate $\sum_{n=1}^{k}e^{in^2}$ and finish through partial summation. 
Details on page $11$ here (it is in Italian, hope you don't mind).
