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arXiv · 2204.00962

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

Gaurav Bhatnagar·Krishnan Rajkumar
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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.

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Gaurav Bhatnagar, Krishnan Rajkumar. 2022-04-03. Telescoping continued fractions for the error term in Stirling's formula. https://arxiv.org/abs/2204.00962

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