Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Abel and Alfred Tauber.

Abelian theorem

A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly.

For example,

implies

for any A(x).

Tauberian theorem

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis which prevents the general term of the series from converging to zero too slowly.