On the coefficients in an asymptotic expansion of (1 + 1∕x)x

Dunster, T. M.; Perez, Jessica M.

doi:10.2140/involve.2021.14.775

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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

DOI: 10.2140/involve.2021.14.775

Abstract

The function $g (x) = {(1 + 1 ∕ x)}^{x}$ has the well-known limit $e$ as $x \to \infty$ . The coefficients $c_{j}$ 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 $| c_{j} | \to 1$ as $j \to \infty$ .

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

Vol. 14, No. 5, 2021