TITLE: When does this series $\sum_{n=0}^{\infty}\sin(n!\pi x)$ converge?
QUESTION [6 upvotes]: My book says that the following series $$\sum_{n=0}^{\infty}\sin(n!\pi x)$$
converges for all rational values of $x$ (which is correct since then all the terms are $0$ after some definite index) and also for $x = e$, $=(2k+1)e$, $=2k/e$, $=\sin1$, $=\cos1$. I don't understand why, since it seems that the general term doesn't go to zero, or am I mistaken? Also is it probably a typo?
For $x = e$, according to Wolfram Alpha, the sum does not converge.

REPLY [6 votes]: 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.