Tweaking Ramanujan’s Approximation of n!

Year 2022, , 10 - 15, 01.03.2022

Sidney Morris

https://doi.org/10.33401/fujma.995150

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.

Keywords

Asymptotic, Gamma function, n!, Ramanujan, Stirling approximation

References

• [1] I. Tweddle, James Stirling’s Methodus Differentialis: An Annotated Translation of Stirling’s Text, Springer, London, 2003.
• [2] R. Michel, The (n+1)th proof of Stirling’s formula, Amer. Math. Monthly, 115 (2008), 844-845, https://doi.org/10.1080/00029890.2008.11920599.
• [3] W. Burnside, A rapidly converging series for logN!, Messenger Math., 46 (1917), 157-159.
• [4] R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA, 75 (1978), 40-42.
• [5] W. D. Smith, The Gamma function revisited, https://schule.bayernport.com/gamma/gamma05.pdf.
• [6] C. Mortici, A substantial improvement of the Stirling formula, Proc. Nat. Acad. Sci. USA, 24 (2011), 1351-1354.
• [7] G. Nemes, On the coefficients of the asymptotic expansion of n!, J. Integer Sequences, 13(6) (2010), Article 10.6.6.
• [8] V. Namias, A simple derivation of Stirling’s asymptotic series, American Math. Monthly, 93 (1986), 25-29.
• [9] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, edited by S. Raghavan and S. S. Rangachari, Springer, New York, 1988.
• [10] M. Hirschhorn, M. B. Villarino, A refinement of Ramanujan’s factorial approximation, Ramanujan J., 34 (2014), 73-81, DOI: 10.1007/s11139-013-9494-y.
• [11] C-P Chen, A more accurate approximation for the Gamma function, J. Number Theory, 164 (2016), 417-428.