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 :=>