Telescoping continued fractions for the error term in Stirling's formula

TL;DR

This paper introduces telescoping continued fractions to derive improved lower bounds for the error term in Stirling's approximation, providing a new method to analyze and bound the approximation error more precisely.

Contribution

It presents a novel telescoping continued fraction technique and an algorithm for bounding the error term in Stirling's formula, enhancing previous bounds.

Findings

Provides tighter lower bounds for the Stirling error term

Develops an algorithm to compute continued fraction terms

Enables experimental bounds for convergents of continued fractions

Abstract

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by Ces\`{a}ro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.