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

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 $j\rightarrow{\infty}$.