TY - JOUR

T1 - A note on the finitization of Abelian and Tauberian theorems

AU - Powell, Thomas

N1 - Funding Information:
This work was supported by the German Science Foundation (DFG Project KO 1737/6‐1). The author is grateful to the anonymous referee for their valuable suggestions, particularly their encouragement to clarify the underlying proof theory.
Publisher Copyright:
© 2020 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/10/7

Y1 - 2020/10/7

N2 - 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 Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.

AB - 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 Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.

UR - http://www.scopus.com/inward/record.url?scp=85091607037&partnerID=8YFLogxK

U2 - 10.1002/malq.201900076

DO - 10.1002/malq.201900076

M3 - Article

AN - SCOPUS:85091607037

SN - 0942-5616

VL - 66

SP - 300

EP - 310

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

IS - 3

ER -