# Does a_n=n^x/(n!)  converge for any x?

May 26, 2016

The sequence ${a}_{n} \to 0$ for any fixed $x$

#### Explanation:

For any fixed $x = {x}_{0}$ there is a ${n}_{0}$ such that for $n > {n}_{0}$ the sequence ${a}_{n} \to 0$. The converse is easy to obtain. Given a ${n}_{0}$ then fixing a x_0 = log_e(n_0!)/log_en_0, for $n > {n}_{0}$ then ${a}_{n} \left({x}_{0}\right) \to 0$

May 26, 2016

For every $x$, lim_(nrarroo)n^x/(n!) = 0

#### Explanation:

Exponential grows faster than power.

$\forall x$ ${\lim}_{n \rightarrow \infty} {n}^{x} / \left({e}^{n}\right) = 0$

Factorial grows faster than exponential.

lim_(nrarroo)e^n/(n!) = 0

The product must also go to $0$