Variations on an error sum function for the convergents of some powers of $e$

TL;DR

This paper proves a previously empirically observed formula involving the sum of errors in convergents of e and extends it to powers of e, providing new insights into their continued fraction approximations.

Contribution

It rigorously proves an empirical formula for the sum of errors in convergents of e and derives similar formulas for powers of e.

Findings

Proved the error sum formula for e's convergents.

Extended formulas to powers of e.

Provided explicit series representations for these sums.

Abstract

Several years ago the second author playing with different "recognizers of real constants", e.g., the LLL algorithm, the Plouffe inverter, etc. found empirically the following formula. Let $p_n/q_n$ denote the $n$th convergent of the continued fraction of the constant $e$, then $$ \sum_{n \geq 0} |q_n e - p_n| = \frac{e}{4} \left(- 1 + 10 \sum_{n \geq 0} \frac{(-1)^n}{(n+1)! (2n^2 + 7n + 3)}\right). $$ The purpose of the present paper is to prove this formula and to give similar formulas for some powers of $e$.