Series: $\Sigma n!/n^n$ - Why Does it Converge?

[badirishluck]

Homework Statement

$$\Sigma$$n!/n^n

index n=1 to infinity

Homework Equations

The Attempt at a Solution

Using the Ratio test (limit as n goes to infinity of a$$_{n+1}$$/a$$_{n}$$)
and found that the series converges.

However, I thought that factorials grew faster than exponential functions. Therefore, it would diverge, right?

Could someone explain why? Did I just do something wrong?

 

[ptr]

Exponents of the form n^n grows much faster than factorials of the form (n!) because the factorial is a multiplication of n terms, the majority of which are less than n, and the power is a multiplication of n terms, all of which are equal to n.

 

[Mark44]

Factorials don't grow faster than exponentials of the sort you're working with. Just think about it: n! = 1 * 2 * 3 * ... * n. You have n factors, of which the largest is n.
n^n = n * n * n * ... * n. Here you have n factors, all of which are n. Clearly this exponental is larger than the factorial above.

 

[Avodyne]

Exponentials with a fixed base, like e^n or 2^n, grow more slowly than n!.