Bounds On The Sum Of A Divergent Series

TL;DR

This paper develops a method using Borel transforms and conformal maps to bound the sum of divergent series, demonstrating its effectiveness on mathematical models and applying it to physical problems like quantum electrodynamics at high temperatures.

Contribution

It introduces a novel approach to bounding divergent series using Borel and conformal techniques, with applications to physical theories and non-perturbative effects.

Findings

Successfully bounds series sums in toy models with known results

Shows non-perturbative contributions clarify Borel-nonsummable series

Provides insights into quantum electrodynamics at extreme conditions

Abstract

Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational conformal map. The method is illustrated by applying it to various mathematical toy-models for which exact results are known. One of these models is used to exemplify how non-perturbative contributions supplement the sum of a Borel-nonsummable series to give the final exact and unambiguous result. Finally, the method is applied to some physical problems. In particular, some speculations are made on the phase of quantum electrodynamics at super-high temperatures from a study of its perturbative free-energy density.