Stirling's approximation is what you need. (Look at the section on "Speed of convergence and error estimates" in that link.)
Prove or Disprove that for any natural number n > 11,
(n^n)/(3^n) < n! < (n^n)/(2^n)
I've tried pretty much everything, and I'm getting absolutely nowhere.
I've broken it into two cases, but can't prove either of them.
At this point I'm not even sure if the statement is actually true and if I should be trying to disprove it instead.
Help would be greatly appreciated. Thanks.