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

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 $p_n\mathrm{/}{q}_n$ denote the $n$th convergent of the continued fraction of the constant $e$, then $$\sum _{n\ge 0}\mathrm{\mid }{q}_{n}e-p_n\mathrm{\mid }=\frac{e}{4}(-1+10\sum _{n\ge 0}\frac{(-1{)}^n}{(n+1)!(2n^2+7n+3)})\mathrm{.}$$ The purpose of the present paper is to prove this formula and to give similar formulas for some powers of $e$.