Abelian and Tauberian theorems

Summary

Abelian and Tauberian theorems are two opposite kinds of results about summability methods. An Abelian theorem says that a method gives the usual value to every convergent series. More generally, it says that summability by one method implies summability by another, with the same value. Abel's theorem is the model. A Tauberian theorem is a partial converse. It says that a summable series converges, provided its terms satisfy an extra Tauberian condition. Tauber's theorem is the model. Tauberian theorems for integral transforms, such as the Wiener–Ikehara theorem, give short proofs of the prime number theorem.