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

TL;DR

This paper investigates the asymptotic expansion coefficients of the function (1+1/x)^x, deriving a recursion, approximating the coefficients for large indices, and showing their magnitude approaches 1 as the index grows.

Contribution

It introduces a simple recursion formula for the coefficients and uses complex analysis to approximate their behavior for large indices, revealing their limiting magnitude.

Findings

Coefficients satisfy a derived recursion formula.

Approximation of coefficients for large j using Cauchy's integral formula.

Magnitude of coefficients approaches 1 as j tends to infinity.

Abstract

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