sequences and series - Convergence of $sum_{k=1}^infty frac{sin(k(k-1))}{k}$

In connection with the strange behavious of a certain sum in Mathematica (https://mathematica.stackexchange.com/q/210849/16361) I suspected a possible divergence but I could not prove of disprove it.

Here's the question: is the sum

$$s_1=\sum_{k=1}^\infty \frac{\sin\left(k(k-1)\right)}{k}$$

convergent or divergent?

Similarly with

$$s_2=\sum_{k=1}^\infty \frac{\sin(k^2)}{k}$$

Numerical evidence (partial sums) seem to indicate convergence.

EDIT 07.12.19

Actually, the story began one step earlier: I considered this unanswered question Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$

$$s_3 =\sum_{k=1}^\infty \frac{\sin(k!)}{k}$$

and wanted to simplify it replacing $k!$ with something simpler.