Elementary Proofs of Some Stirling Bounds

Abstract: 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 $n\ge 8$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n}-\frac{1}{360n^3+103n}}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e<sup>{\frac{1}{12n}-\frac{1}{360n<sup>3+102n}}$$ and for all $n\ge 3$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e^{{\frac{1}{12n+\frac{2}{5n}-\frac{1.1}{10n^3}}}} \ge n!\ge \sqrt{2\pi n}\left(\frac{n}{e}\right)ⁿ e<sup>{\frac{1}{12n+\frac{2}{5n}-\frac{0.9}{10n<sup>3}}}.$$