Question

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

Answer 1

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!)|`

`=|((n+1))/((n+1)^(n+1))*n^n|`

`=|(n^n)/(n+1)^n|`

Therefore,

`lim_(n->oo)|a_(n+1)/a_n|=lim_(n->oo)|(n^n)/(n+1)^n|`

`=lim_(n->oo)(n/(n+1))^n`

`=lim_(n->oo)((n+1)/(n))^-n`

`=1/e`

As

`lim_(n->oo)|a_(n+1)/a_n| <1`

The series converges.