A simple proof of monotonicity for remainder of Stirling's formula

TL;DR

This paper presents a straightforward proof demonstrating the monotonicity of the remainder term in Stirling's approximation for the gamma function, utilizing integral transforms with series.

Contribution

It provides a simple, clear proof of the monotonicity property, enhancing understanding of Stirling's formula remainder behavior.

Findings

Confirmed monotonicity of the Stirling remainder term.

Simplified proof method using integral transforms.

Improved theoretical understanding of Stirling's approximation.

Abstract

The monotonicity properties of remainder of Stirling's formula for the gamma function are simply obtained by using the integral transforms with series.