When does this series $\sum_{n=0}^{\infty}\sin(n!\pi x)$ converge?

Idea:

$$n!e = n!(1+1/1!+1/2!+1/3! +\cdots+1/n! + r_n).$$

Verify that $n!(1+1/1!+1/2!+1/3! +\cdots+1/n!)$ is an odd integer if $n$ is even, and is an even integer if $n$ is odd. Also, $r_n$ decreases to $0$ as $n\to \infty.$ So I think $\sum \sin(n!\pi e)$ is an alternating series whose terms in absolute value decrease to $0.$ Hence this series converges.

I think I can prove it for the first one, and then the others should be similar. In the case of $$\sum_{n=0}^\infty\sin(n!\pi e)$$ we need to consider the value of $n!\pi e\operatorname{mod} \pi$, and it's enough to consider the value of $n!e\operatorname{mod}1$. Now $$n!e=n!\sum_{k=0}^\infty\frac1{k!}$$ so that $$n!e\operatorname{mod} 1=n!\sum_{k=n+1}^\infty\frac1{k!}<\sum_{k=1}^\infty\frac1{(n+1)^k}=\frac1n\to0\text{ as } n\to\infty$$