A Fun Limit: relationship between (n!)^(1/n) and n

We all know that \(n^n >> n! >> a^n >> n^a >> n\), but what about \((n!)^{\frac{1}{n}}\) and \(n\)?

Proof

A lemma

The proof of

\[ \liminf _{n\rightarrow \infty }\dfrac{a_{n+1}}{a_{n}}\leq \liminf _{n\rightarrow \infty }\sqrt[n] {a_n}\leq \limsup _{n\rightarrow \infty }\sqrt[n] {a_n}\leq \limsup _{n\rightarrow \infty }\dfrac{a_{n}+1}{a_n} \]

Can be found here