  We give elementary proofs of several Stirling's precise bounds. We first
improve all the precise bounds from the literature and give new precise bounds.
In particular, we show that for all $n\ge 8$ $$\sqrt{2\pi
n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}-\frac{1}{360n^3+103n}} \ge n!\ge
\sqrt{2\pi n}\left(\frac{n}{e}\right)^n
e^{\frac{1}{12n}-\frac{1}{360n^3+102n}}$$ and for all $n\ge 3$ $$\sqrt{2\pi
n}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+\frac{2}{5n}-\frac{1.1}{10n^3}}}
\ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)^n
e^{\frac{1}{12n+\frac{2}{5n}-\frac{0.9}{10n^3}}}.$$

Functional Analysis;Computational Complexity