10
$\begingroup$

I'd love your help this time with the following limit:

$\lim_{n \to +\infty } \left \{ en! \right \}$

when $\{ a \}=a-[a].$

Honestly, I don't have a clue.

Thank you.

  • 3
    @Alon,@Jonas: Do you mean $e=\sum_{k=0}^{\infty}\frac{1}{k!}$?2011-06-19

2 Answers 2

14

You have $n!e=n!(1+\frac{1}{2!}+...+\frac{1}{n!})+\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+....+\frac{1}{(n+1)(n+2)....(n+p)}+...$.

We have $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+....+\frac{1}{(n+1)(n+2)....(n+p)}+...$ $\leq \frac{1}{n+1}+\frac{1}{(n+1)^2}+...+\frac{1}{(n+1)^p}+...=$ $=\frac{1}{n+1}(1+\frac{1}{n+1}+...+\frac{1}{(n+1)^p}+..)=\frac{1}{n+1}\frac{1}{1-\frac{1}{n+1}}=\frac{1}{n}$.

This means that $\{n!e\}\leq \frac{1}{n}$, and the conclusion follows.

3

Use the fact that $ en! = n! \cdot\biggl(1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} \biggr) + n! \cdot \biggl( \frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \cdots \biggr)$