Converting a series in \lambda to a series in \lambda^{-1}

TL;DR

This paper presents a transformation that converts series in a parameter  into series in , revealing relations between convergent and divergent series and aiding in the analysis of large- and small-coupling expansions.

Contribution

It introduces a novel transformation method for converting series in  to series in , providing insights into their relationships and applications to divergent series like the Euler-Heisenberg-Schwinger expansion.

Findings

The transform links convergent and divergent series.

Application to Euler-Heisenberg-Schwinger series demonstrates its utility.

Provides a new perspective on series summation and analytical continuation.

Abstract

We introduce a transformation for converting a series in a parameter, \lambda, to a series in the inverse of the parameter \lambda^{-1}. By applying the transform on simple examples, it becomes apparent that there exist relations between convergent and divergent series, and also between large- and small-coupling expansions. The method is also applied to the divergent series expansion of Euler-Heisenberg-Schwinger result for the one-loop effective action for constant background magnetic (or electric) field. The transform may help us gain some insight about the nature of both divergent (Borel or non-Borel summable series) and convergent series and their relationship, and how both could be used for analytical and numerical calculations.