The nth root of n! (n factorial)

This page gives a lower bound for the nth root of the factorial of \(n\), \(n!\).

\begin{align} e^n n! &= \sum_{i=1}^\infty \frac{n^i}{i!}n! \\ &= n^n+n!\sum_{i=1,i\neq n}^\infty\frac{n^i}{i!} \end{align}

All the terms in the summation after \(n^n\) are positive, so \[ e^n n!>n^n \] and so \[ n!>(n/e)^n \] and \[ (n!)^{1/n}>n/e. \]

Thus \((n!)^{1/n}\) goes to infinity as \(n\) increases.