Show that $\{\frac{e^n}{n!}\}$ diverge.

Question

I know that the sequence $\{\frac{e^n}{n!}\}$ diverges and that for prove it i have to limit it, but i don't know how do it.
In fact, i know that $\{\frac{x^n}{n!}\}$ diverges, but i don't know prove it.

Answer 1

Notice $\sum \frac{ x^n }{n!} $ converges for all $x$. In particular for $x=e$. Thus,

$$ \lim_{n \to \infty} \frac{e^n}{n!} = 0 $$

Answer 2

Note that by ratio test

$$\frac{e^{n+1}}{(n+1)!}\frac{n!}{e^n}=\frac{e}{n+1}\to 0<1$$

then

$$\frac{e^n}{n!}\to 0$$