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A note on the finitization of Abelian and Tauberian theorems
Author(s) -
Powell Thomas
Publication year - 2020
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201900076
Subject(s) - mathematics , finitary , abelian and tauberian theorems , converse , limit (mathematics) , abelian group , interpretation (philosophy) , simple (philosophy) , series (stratigraphy) , pure mathematics , calculus (dental) , discrete mathematics , mathematical analysis , epistemology , medicine , dentistry , paleontology , philosophy , geometry , computer science , biology , programming language
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.