What is larger for large n: $(n!)^{n^n}$ or $(n^n)!$?

Question

For large enough n, what is larger: $(n!)^{n^n}$ or $(n^n)!$? I’ve been given this problem in my calculus class and I have no idea how to approach it. All help is welcome.

Answer 1

From Stirling formula, we know $\ln n! = n \ln n - n + O(\ln n)$

Taking logarithm of both expressions, we need to compare $$n^n \cdot \ln n! = n^{n + 1} \ln n - n^{n + 1} + O(n^n \ln n) < n^{n + 1} \ln n - \frac{1}{2} \cdot n^{n + 1}$$ and $$n^n \cdot (\ln n^n) - n^n + O(\ln n^n) = n^{n + 1} \ln n - n^n + O(\ln n^n) > n^{n + 1}\ln n - 2 n^n > n^{n+1}\ln n - \frac{1}{2}\cdot n^{n + 1}$$