Stirling's Formula/Refinement

Theorem
A refinement of [Stirling's Formula]] is:
 * $n! \sim \sqrt {2 \pi n} \left({\dfrac n e}\right)^n \left({1 + \dfrac 1 {12 n} }\right)$

where $\sim$ denotes asymptotically equal.

Proof
From Limit of Error in Stirling's Formula:


 * $e^{1 / \left({12 n + 1}\right)} \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$