Question

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

Answer 1

The series converges

Explanation:

The `n`-th term of the series is `t_n = (n!)/n^n`. Hence we have

`t_(n+1)/t_n = ((n+1)!)/(n+1)^(n+1) times n^n/(n!)`
`qquad = (n+1)/(n+1)^(n+1) n^n = (n/(n+1))^n`
`qquad = 1/(1+1/n)^n`

Now, it is well known that `lim_{n to oo} (1+1/n)^n = e` (indeed, that's the definition of the number `e`). And thus

`lim_{n to oo }|t_{n+1}/t_n| = 1/e <1`

and thus the series converges according to the ratio test.