Convergency of $\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}$ I am stuck on how to prove the convergency of the series $$\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}.$$ It seems like that the series converges to approximately $2.85$, but I have no idea how to show whether the series converges or not. I know that there are techniques such as the ratio test, the integral test, the $n^{th}$ term test, but I still do not know how to approach the problem. Could you give me a hint?

$\pi\
ot\in\mathbb{Q}$ has a finite irrationality measure, in particular there are a finite number of $\frac{p}{q}\in\mathbb{Q}$ such that $$ \left|\pi - \frac{p}{q}\right|\leq \frac{1}{q^{10}}\quad\Longleftrightarrow\quad d(p,\pi\mathbb{Z})\leq\frac{1}{q^9} $$ and for any sufficiently large $n\in\mathbb{N}^+$ we have $$ \left|\sin(n)\right|\geq \tfrac{2}{\pi}d(n,\pi\mathbb{Z})\geq 2\pi^8\cdot\frac{1}{n^9}. $$ Since the series $\sum_{n\geq 1}\frac{n^9}{n!}$ is convergent, the series $\sum_{n\geq 1}\frac{1}{n!\sin(n)}$ is absolutely convergent.