Tauberian condition

Summary

A Tauberian condition is an extra hypothesis on the terms of a series that lets one pass from summability back to ordinary convergence. Abel's theorem says that a convergent series is Abel summable to its sum, but the converse fails. Tauber's theorem shows that the converse holds when $n a_n \to 0$, and this condition is the first example of a Tauberian condition. Such conditions are the subject of Tauberian theory.