# Convergence from Divergence

## TL;DR

This paper introduces a method to transform divergent series common in physics into rapidly convergent inverse factorial series, enabling rigorous extrapolation across different parameter regimes and connecting to existing summation techniques.

## Contribution

The authors develop a novel resummation technique converting divergent series into convergent inverse factorial series, enhancing the analysis of asymptotic expansions in physics.

## Key findings

- Convergent inverse factorial series can replace divergent perturbative series.
- The method allows rigorous extrapolation from large to small parameters.
- Connections to Borel summation and the Stokes phenomenon are established.

## Abstract

We show how to convert divergent series, which typically occur in many applications in physics, into rapidly convergent inverse factorial series. This can be interpreted physically as a novel resummation of perturbative series. Being convergent, these new series allow rigorous extrapolation from an asymptotic region with a large parameter, to the opposite region where the parameter is small. We illustrate the method with various physical examples, and discuss how these convergent series relate to standard methods such as Borel summation, and also how they incorporate the physical Stokes phenomenon. We comment on the relation of these results to Dyson's physical argument for the divergence of perturbation theory. This approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions, and Klein-Gordon and Dirac determinants.