Stirling's Formula/Refinement

Theorem
A refinement of Stirling's Formula is:
 * $n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$

where $\map \OO \cdot$ is the big-$\OO$ notation.

Proof
We have:
 * $\forall x \in \closedint 0 1 : \dfrac {x^2} 2 \le e^x - \paren {1 + x} \le e x^2$

since:

Thus the claim follows from Limit of Error in Stirling's Formula:
 * $e^{1 / \paren {12 n + 1} } \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$