What is $O\Big((n+1)!\Big)$?

Question

What is $f(n) = (n+1)!$ which is also $f(n) = (n+1)n!$ in terms of big-O notation? My guess is $O(n \cdot n!)$ but I am not sure. I only know it is certainly $f(n) \in O(n^n)$.

Answer 1

Using Stirling expansion for $n! \sim \sqrt{2 \pi}(\frac{n}{e})^{n} n^{\frac{1}{2}}$ your expression is $n!n +O(n!)$. Hence $(n+1)! \sim \sqrt{2 \pi} n^{n+\frac{3}{2}} e^{-n} +O(n!)$