Abel summation

Summary

Abel summation is the summability method that assigns to a series $\sum a_n$ the limit of the power series $\sum a_n x^n$ as $x \to 1$ from below. By Abel's theorem, it gives the usual sum to every convergent series. It sums every series that Cesàro summation sums, to the same value, and more. Its partial converses, such as Tauber's theorem and Littlewood's Tauberian theorem, started the theory of Tauberian theorems.