Tweaking Ramanujan’s Approximation of n!

Authors : Sidney MORRİS

Pages : 10-15

Doi:10.33401/fujma.995150

View : 23 | Download : 10

Publication Date : 2022-03-01

Article Type : Research Paper

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 insert ignore into journalissuearticles values(i); how all these asymptotic results can be easily verified; insert ignore into journalissuearticles values(ii); how Hirschhorn and Villarino`s argument allows tweaking of Ramanujan`s result to give a better approximation; and insert ignore into journalissuearticles values(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.
Keywords : Asymptotic, Gamma function, n, Ramanujan, Stirling approximation