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

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 $$\sum _{n\ge 0}|q_{n}e-p_n|=\frac{e}{4}\left(-1+10\sum _{n\ge 0}\frac{(-1{)}^n}{(n+1)!(2n^2+7n+3)}\right).$$ The purpose of the present paper is to prove this formula and to give similar formulas for some powers of $e$.