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