Tweaking Ramanujan's Approximation of n!

TL;DR

This paper reviews and compares various asymptotic formulas for factorials, highlighting Ramanujan's improved approximation, and introduces new tweaks and comparisons to enhance factorial approximation accuracy.

Contribution

It demonstrates how existing asymptotic results can be verified, improved, and compared, including deriving a new asymptotic formula close to Chen's.

Findings

Ramanujan's approximation can be tweaked for better accuracy

A new asymptotic formula is derived with improved precision

Chen's formula outperforms other known asymptotic approximations

Abstract

In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows a tweaking of Ramanujan's result to give a better approximation; (iii) that a new asymptotic formula can be obtained by further tweaking of Ramanujan's result; (iv) that Chen's asymptotic formula is better than the others mentioned here, and the new asymptotic formula is comparable with Chen's.