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

2022/04/03 by Gaurav Bhatnagar, Bhatnagar, Gaurav, Krishnan Rajkumar +1
Mathematics · #11J70 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Functional Equations Stability Results #Iterative Methods for Nonlinear Equations #Number Theory (math.NT) #Primary: 33C45 #Secondary: 30B70

paper · pdf · doi:10.48550/arxiv.2204.00962

openalex publication_date 2022/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term r_n in Stirling's approximation n! = √(2π)n^n+1/2e^-ne^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.

Related

• Stirling's formula derived simply
• Variations on an error sum function for the convergents of some powers of e
• Elementary Proofs of Some Stirling Bounds
• Original proofs of Stirling's series for log(n!)
• Tweaking Ramanujan's Approximation of n!