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

Abstract

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $n!=\sqrt{2\pi }{n}^{n+1/2}{e}^{-n}{e}^{r_n}.$ This improves lower bounds given earlier by Cesàro (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.

Final version submitted to the Journal of Approximation Theory accepted for publication. Please see ver 1 for partial proofs of some observations and more discussion of the conjectures mentioned here