What is the Limit of the nth Root of n Factorial as n Approaches Infinity?

[foo_daemon]

Homework Statement

Prove the \lim_{n\to +\infty}{\sqrt[n]{ n! }} \equiv \infty

Homework Equations

uses well-known operations

The Attempt at a Solution

I think the best (easiest) approach is to find some f(n) \leq n! , and express it as some g(n)^{n} . This will then get rid of the annoying n-root, and it should then be easy to show that \lim_{n\to +\infty}{g(n)} = \infty , which implies the limit is infinity for n! (i.e. the squeeze theorem only with regard to the lower bound).

Since n! = (n)(n-1)(n-2) ... (n-n+2) , I thought of using the smallest multiple ( n - n + 2)^{n-1}, but I still cannot express this as a function to the n power, and even if I did, the limit would just be 2.. so it's smaller than n!, but not 'big enough'.

I think I may need a different approach. Suggestions?

 

[Outlined]

Would try to use r = \exp(\ln(r)). This gives

\prod_{i = 1}^{n}\exp(\frac{\ln i}{n})