A Short Proof of Stirling's Formula

TL;DR

This paper presents a concise proof of Stirling's formula and its refinement by employing variable substitution and dominated convergence techniques, simplifying the derivation process.

Contribution

It introduces a novel, streamlined proof of Stirling's formula using advanced calculus methods, improving clarity and brevity over traditional proofs.

Findings

Provides a shorter proof of Stirling's formula

Includes a refinement of Stirling's approximation

Uses dominated convergence for rigorous justification

Abstract

By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.