Let $\epsilon_n$ be the fractional party of $n!e$, where $n$ is a positive integer. i. Show that $\frac1{n+1}<\epsilon_n<\frac1n$ for all positive integers $n$. ii. Prove that $n\sin(2n!e\pi)\rightarrow 2\pi$ as $n\to\infty$
ii can be shown using i and the fact that $\frac{\sin x}x\to1$ as $x\to0$, taking $x=\frac{2\pi}n$. In proving i, one can easily show $\frac1{n+1}<\epsilon_n$ by using the fact $\epsilon_n=\frac1{n+1}+\frac1{(n+1)(n+2)}+\frac1{(n+1)(n+2)(n+3)}+\cdots$. But how to prove $\epsilon_n<\frac1n$?
$$\epsilon_n<\frac{1}{n+1}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots.$$ What does this series converge to?