TITLE: How do I prove convergence of $\sum\limits_{k=1}^\infty \tfrac{\sin k^2}{k}$?
QUESTION [5 upvotes]: How do I prove convergence of $\sum\limits_{k=1}^\infty \tfrac{\sin k^2}{k}$?
I would prefer avoid using Taylor expansion...

REPLY [4 votes]: It is not easy. The convergence of the more general series
$$
\sum_{n=1}^\infty\frac{\sin(n^k)}{n},\quad k>0,
$$
is studied in this question.