# How do you use the Ratio Test on the series sum_(n=1)^oon^n/(n!) ?

Oct 7, 2014

Recall: The Definition of Number $e$

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

(Note: This can be derived suing l'Hopital's Rule as well.)

Now, let us examine the convergence of the posted series.

By the Ratio Test ,

lim_{n to infty}|{a_{n+1}}/{a_{n}}|=lim_{n to infty}|(n+1)^{n+1}/{(n+1)!}cdot{n!}/{n^n}|

by cancelling out common factors,

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

by simplifying a bit further,

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

Hence, the series diverges.

I hope that this was helpful.