A note on the finitization of Abelian and Tauberian theorems

TL;DR

This paper provides finitary, proof-theoretic formulations of Abel's and Tauber's theorems, offering quantitative versions of these classical results on infinite series convergence.

Contribution

It introduces a novel proof-theoretic approach inspired by G"{o}del's functional interpretation to finitize and quantify these theorems.

Findings

Quantitative versions of Abel's and Tauber's theorems derived.

Finitary formulations enable computational interpretations.

Proof-theoretic methods applied to classical analysis results.

Abstract

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular G\"{o}del's functional interpretation, which we use to establish quantitative version of both of these results.