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

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