  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.
