Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials

TL;DR

This paper provides a detailed analysis of the truncation errors of the Euler series, deriving asymptotic expansions using exponential polynomials and Laguerre polynomials, and examines their convergence properties.

Contribution

It introduces novel asymptotic expansions of the Euler series remainder expressed through exponential and Laguerre polynomials, enhancing understanding of their divergence and convergence.

Findings

Explicit closed-form coefficients in terms of exponential polynomials.

Asymptotic expansions in inverse powers and factorials of n.

Analysis of convergence and divergence for positive arguments.

Abstract

A detailed analysis of the remainder obtained by truncating the Euler series up to the $n$th-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse powers and in inverse rising factorials of $n$, are obtained. It is found that the corresponding expanding coefficients are expressed, in closed form, in terms of exponential polynomials, well known in combinatorics, and in terms of associated Laguerre polynomials, respectively. A study of the divergence and/or of the convergence of the above expansions is also carried out for positive values of the Euler series argument.