Elementary Proofs of Some Stirling Bounds

TL;DR

This paper provides elementary proofs of improved and new precise bounds for Stirling's approximation of factorials, enhancing accuracy for all sufficiently large n with explicit exponential correction terms.

Contribution

It introduces elementary proofs that tighten existing Stirling bounds and presents new bounds with explicit correction factors for large n.

Findings

Improved Stirling bounds for n ≥ 8.

New precise bounds for n ≥ 3.

Explicit exponential correction terms enhance approximation accuracy.

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