Vol. 14, No. 5, 2021

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On the coefficients in an asymptotic expansion of $(1+1/x)^x$

T. M. Dunster and Jessica M. Perez

Vol. 14 (2021), No. 5, 775–781
Abstract

The function g(x) = (1 + 1x)x has the well-known limit e as x . The coefficients cj in an asymptotic expansion for g(x) are considered. A simple recursion formula is derived, and then using Cauchy’s integral formula the coefficients are approximated for large j. From this it is shown that |cj| 1 as j .

Keywords
series expansions, asymptotic representations in the complex plane, integrals of Cauchy type
Mathematical Subject Classification
Primary: 41A58, 30E15, 30E20
Milestones
Received: 12 May 2020
Accepted: 8 May 2021
Published: 9 February 2022

Communicated by Kenneth S. Berenhaut
Authors
T. M. Dunster
Department of Mathematics and Statistics
San Diego State University
San Diego, CA
United States
Jessica M. Perez
Department of Mathematics and Statistics
San Diego State University
San Diego, CA
United States