Krishnan Rajkumar (JNU) and I have a new preprint -- on telescoping continued fractions. I have written previously on telescoping and several times on continued fractions, but this one is unique. I don't think anyone has tried to combine the two ideas. We apply it to find lower bounds for the remainder term for Stirling's formula. Ultimately, we discovered a nice new technique, proved several things required to make it work, but were unable to take it to its natural conclusion (so far!). The preprint (now available at arxiv) has several conjectures.

Here is a link to the preprint.

Here is the abstract:

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

Authors: Gaurav Bhatnagar, Krishnan Rajkumar

Categories: math.CA math.NT

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.

Here is a talk introducing the method. I presented in the Ashoka Math Colloquium on November 2, 2021. It has an overview of the technique. The talk was made keeping undergraduate students in mind, so there is something here which is quite accessible. In particular I have outlined Robbins' approach at the beginning of the talk.