I have to prove and exhibit where the following succession tends, but there are some issues, I tried to use Taylor or the Stirling formula, because it seemed to me that I could re-write the terms more easily, but both failed. How should I proceed?
Thanks.
$a_n = \sin( 2\pi en! )$
($n\in\mathbb{N}$)
Using Taylor series, $e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} + \cdots$. Therefore, $en! = n! + n! + \frac{n!}{2!} + \cdots + 1 + \frac{1}{n + 1} + \cdots$. All of the terms up until that $1$ are integers, so $en!$ is within $\frac{1}{n + 1} + \frac{1}{(n + 1)(n + 2)} + \cdots \leq \frac{2}{n}$ of an integer (that's just the most convenient upper bound I could think of right away).
Can you finish it from there?