Self-Similar Bootstrap of Divergent Series

TL;DR

The paper introduces a self-similar bootstrap method for summing divergent series, utilizing algebraic transformations to enhance stability and enable complete renormalization, demonstrated through examples in statistical physics.

Contribution

It presents a novel self-similar bootstrap technique that improves the summation of divergent series using algebraic transformations for maximal stability.

Findings

Effective sums of divergent series can be calculated using the method.

The approach is demonstrated with examples from statistical physics.

The method achieves complete renormalization of series.

Abstract

A method is developed for calculating effective sums of divergent series. This approach is a variant of the self-similar approximation theory. The novelty here is in using an algebraic transformation with a power providing the maximal stability of the self-similar renormalization procedure. The latter is to be repeated as many times as it is necessary in order to convert into closed self-similar expressions all sums from the series considered. This multiple and complete renormalization is called self-similar bootstrap. The method is illustrated by several examples from statistical physics.