Why does (n^n)/(n!) grow faster than any finite bound?

[burritoloco]

Homework Statement

Determine the limit of the sequence (n^n)/(n!)

Homework Equations

The Attempt at a Solution

I think the limit should be infinity as n^n grows faster than n!, but I'm not sure how to prove it. Thanks for the help!

 

[LCKurtz]

Write it as

$$\frac n n\cdot \frac{n}{n-1}\cdot\frac{n}{n-2}\cdots\frac n 2\cdot\frac n 1$$

and see if you can underestimate it with something going to infinity.

 

[SammyS]

Can you show that (n+1)⁽ⁿ⁺¹⁾/(n+1)! > nⁿ/n! for n>1 ?

This won't compete the proof, but it's a start.