Simpson's rule for estimating n! (and proving Stirling's formula, almost)

Abstract

Using a tool familiar to first-year calculus students (Simpson's rule) surprisingly good estimates are deduced for values of n!—or more precisely ln (n!)—along with error bounds. These estimates can be implemented on a simple hand-held calculator or computer. Moreover, it is demonstrated how to arrive at analogous, improved estimates (with error bounds) for all higher-order Newton-Cotes integration methods. Along the way, the error bounding naturally, and in short order, leads to the conclusion that n! ∼ C(n/e)_n √n. While these methods cannot show the entirety of Stirling's formula (namely that C = √2φ), they do show how C can be approximated to any desired accuracy.