# Proof of lim (n!)^(1/n)

## 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?

Would try to use $$r = \exp(\ln(r))$$. This gives
$$\prod_{i = 1}^{n}\exp(\frac{\ln i}{n})$$