How Do I Prove That the Limit of \( n!^{n^{-n}} \) Equals 1?

[bomba923]

How do I show that \mathop {\lim }\limits_{n \to \infty } \left( {n!} \right)^{\left( {n^{ - n} } \right)} = 1 ?

(The n^-n forces the value to decrease faster than n! increases, I believe. But how to work out that?)

 

[estalniath]

1)Firstly we'll let the expression (n!)^(n^-n)=y, then taking ln on both sides give,
lny=ln(n!)/n^n.
2) We'll then find the limit of the expression at the RHS using Le Hopital's rule or by observation that n^n increases faster than ln(n!). =)
3) After finding the limit, L, all we have to do is substitute back the value of y, which is e^L