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

1 Answer
Oct 7, 2014

Recall: The Definition of Number e

e=lim_{n to infty}(1+1/n)^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}{(n+1)^n}/{n^n}

by simplifying a bit further,

=lim_{n to infty}(1+1/n)^n=e ge 1

Hence, the series diverges.

I hope that this was helpful.