Abelian and Tauberian Theorems

In mathematics, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general abelian and Tauberian theorems give similar results for more general summation methods.

There is no clear distinction between abelian and Tauberian theorems, or even a generally accepted definition of what these terms mean. Often, a theorem is called "abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "tauberian" if it gives conditions for a series summable by some method to be summable in the usual sense.