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

Gaurav Bhatnagar; Krishnan Rajkumar

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

Classical Analysis and ODEs 2023-07-03 v2 Number Theory

Authors: Gaurav Bhatnagar , Krishnan Rajkumar

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.

Keywords

continued fractions bound upper bounds

Cite

@article{arxiv.2204.00962,
  title  = {Telescoping continued fractions for the error term in Stirling's formula},
  author = {Gaurav Bhatnagar and Krishnan Rajkumar},
  journal= {arXiv preprint arXiv:2204.00962},
  year   = {2023}
}

Comments

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

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

Abstract

Keywords

Cite

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