Limit of Error in Stirling's Formula

Theorem
Consider Stirling's Formula:
 * $n! \sim \sqrt {2 \pi n} \left({\dfrac n e}\right)^n$

The ratio of $n!$ to its approximation $\sqrt {2 \pi n} \left({\dfrac n e}\right)^n$ is bounded as follows:
 * $e^{1 / \left({12 n + 1}\right)} \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$

Proof
Let $d_n = \ln n! - \left({n + \dfrac 1 2}\right) \ln n + n$.

From the argument in Stirling's Formula: Proof 2 we have that $\left\langle{d_n - \dfrac 1 {12 n} }\right\rangle$ is an increasing sequence.