A Sterling Look at Stirling’s Constant

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Acknowledgments

The authors wish to thank an anonymous referee for tremendously helpful suggestions, which significantly improved the article’s simplicity and readability.

Summary

In a previous article [3] appearing in this journal, existence of the finite positive limit $\kappa =\underset{n\to \infty }{\lim }n!{(\frac{e}{n})}^n{n}^{-1/2}$ implicit in Stirling’s approximation $n!\approx \sqrt{2\pi n}{(\frac{n}{e})}^n$ was proved, and in so doing this limit was connected to the Euler limit $\underset{n\to \infty }{\lim }{(1+\frac{1}{n})}^n=e$. Because it was not done in the original article, exact determination of the constant $\kappa$ ($=\sqrt{2\pi }$) by a novel, elementary technique is the aim of this follow-up. For both articles, the arguments are readily accessible to advanced undergraduate students with a year of university calculus in their background.

Additional information

Notes on contributors

Adam Hammett

Adam Hammett ([email protected], MR ID 840691, ORCID 0000-0002-1070-0807) earned his Ph.D. in combinatorial probability from The Ohio State University in 2007. Ever since he has taught at the college level, and currently serves as professor of mathematics at Cedarville University, where he has been since 2015. He enjoys overseeing undergraduate research projects, reading, and spending time outdoors with his wife Rachael and their four children Isabelle, Madison, Daniel, and Esther.

Kevin Roper

Kevin Roper ([email protected], MR ID 363586) is originally from Jamaica, and earned his Ph.D. in function theory in several complex variables from the University of Kentucky in 1995. Currently he serves as professor of mathematics at Cedarville University. When not discussing mathematics with his students you can find him on the soccer field coaching the goalkeepers of the university’s womens soccer team or at the firehouse where he is a volunteer firefighter. He and his wife Lynn enjoy getting away to explore new places.