MHB Fundamental theorem and limit proofs

[devorahstar]

Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero.

The hint is to use Stirling's approximation. What is this?

 

[Nono713]

Take the logarithm of the limit, and pull the logarithm inside the limit by using the continuity of the logarithm function. What are you left with? Can you approximate $\log(n!)$ using Stirling's approximation? Simplify, and then you should be able to conclude (don't forget to undo the logarithm you applied to the limit at the start).