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Tweaking Ramanujan’s Approximation of n!

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posted on 2024-07-28, 23:46 authored by Sidney MorrisSidney Morris

Abstract: About 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 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 tweaking of Ramanujan's result to give a better approximation; and (iii) that new asymptotic formulae can be obtained by further tweaking of Ramanujan's result. Tables are calculated displaying how good each of these approximations is for $n$ up to one million.

History

Publication Date

2022-11-01

Journal

Fundamental Journal of Mathematics and Applications

Volume

5

Issue

1

Pagination

6p. (p. 10-15)

Publisher

Fuat USTA

ISSN

2645-8845

Rights Statement

© 2022 The Author. The published articles in Fundamental Journal of Mathematics and Applications are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. The Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license allows the work to be shared, copied, and reproduced in any size and format except for commercial use. It also allows it to be rearranged, transformed, and adapted, including building upon the original work, with proper reference to it.

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