calculus - Convergence of $,,sum_{k=1}^infty frac{sin(k)}{k!}$

The task is to determine if the series converges absolutely, conditionally or doesn't converge at all.

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

I have tried solving it with D'Alembert test and comparison test method. No luck.
We haven't covered integration of

$$\int_1^\infty \frac{\sin(k)}{k}$$

I am stuck. Please give me a hint how to solve it.

Thank you for your attention!

Answer

First we look at the absolutevalue of the series. You know that the $\sum_{n=1}^{ \infty} \dfrac {1}{n^2}$ converges, so to prove convergence (using the comparison test) it is enough to prove that for large enough $n$, $n(n-1)(n-2)>n^2$

$n(n^2-3n+2)>n^2$

$n^2-4n+2>0$ This is clearly true for large enough $n$.

If the absolute value of the series converges, then the original one does also.