A note on the finitization of Abelian and Tauberian theorems

摘要

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 underwhich the converse holds.Our approach is inspired by proof theory,and in particular G?del's functional interpretation,which we use to establish quantitative versions of both of these results.