# How do you show the convergence of the series (n!)/(n^n) from n=1 to infinity??

May 16, 2018

The series converges

#### Explanation:

The $n$-th term of the series is t_n = (n!)/n^n. Hence we have

t_(n+1)/t_n = ((n+1)!)/(n+1)^(n+1) times n^n/(n!)
$q \quad = \frac{n + 1}{n + 1} ^ \left(n + 1\right) {n}^{n} = {\left(\frac{n}{n + 1}\right)}^{n}$
$q \quad = \frac{1}{1 + \frac{1}{n}} ^ n$

Now, it is well known that ${\lim}_{n \to \infty} {\left(1 + \frac{1}{n}\right)}^{n} = e$ (indeed, that's the definition of the number $e$). And thus

${\lim}_{n \to \infty} | {t}_{n + 1} / {t}_{n} | = \frac{1}{e} < 1$

and thus the series converges according to the ratio test.