Lim n->inf, (n!)^(1/n)=?

I have two solutions about this problem.

The easier one is like this:

We can say for any positive real number k,

lim n->inf k^n/n!=0<1

This is because k is a finite number, and so trivial. Thus,

k^n<n!

k<(n!)^(1/n)

And we said that k is every real number. So,

lim n->inf (n!)^(1/n) is greater than any real number. As a result, the value diverge.

For The harder one, we'll use Gamma fuction. I won't explain that because it is a little bit long, but the main idea is to take exp(ln) to the limit and transform n! to the Gamma(n+1). Plus, use L'Hopital's rule, and then we get some digamma fuction. Finally the limit changes into limit n->inf exp(digamma(n+1)), whose value diverges.

Since Im not Amercian, the writing might be unfluent :=>