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 $x\to \infty$. 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 $j\to \infty$.