Question

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

Answer 1

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.