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

Aug 5, 2018

The series converges.

#### Explanation:

The term of the series is

a_n=(n!)/(n^n)

The ratio test is

|a_(n+1)/a_n|=|((n+1)!)/((n+1)^(n+1))*n^n/(n!)|

$= | \frac{\left(n + 1\right)}{{\left(n + 1\right)}^{n + 1}} \cdot {n}^{n} |$

$= | \frac{{n}^{n}}{n + 1} ^ n |$

Therefore,

${\lim}_{n \to \infty} | {a}_{n + 1} / {a}_{n} | = {\lim}_{n \to \infty} | \frac{{n}^{n}}{n + 1} ^ n |$

$= {\lim}_{n \to \infty} {\left(\frac{n}{n + 1}\right)}^{n}$

$= {\lim}_{n \to \infty} {\left(\frac{n + 1}{n}\right)}^{-} n$

$= \frac{1}{e}$

As

${\lim}_{n \to \infty} | {a}_{n + 1} / {a}_{n} | < 1$

The series converges.