Original proofs of Stirling's series for log(n!)

TL;DR

This paper transcribes Stirling and De Moivre's original derivations of the asymptotic series for log(n!), discusses Wallis's infinite product for pi, and clarifies Stirling's priority over De Moivre.

Contribution

It provides the first modern transcription of Stirling's and De Moivre's original proofs of Stirling's series for log(n!), highlighting historical priority.

Findings

Stirling's original derivation of the asymptotic series is clarified.

Wallis's infinite product for pi is discussed in relation to Stirling's series.

The divergence of Stirling's series is analyzed.

Abstract

Transcription into modern notations of the derivation by Stirling and De Moivre of an asymptotic series for $\log(n!)$, usually called Stirling's series. The previous discovery by Wallis of an infinite product for $\pi$, and later results on the divergence of the series are also presented. We conclude that James Stirling has priority over Abraham de Moivre for Stirling's formula and Stirling's series.