Abelian Means
Suppose λ = {λ0, λ1, λ2, ...} is a strictly increasing sequence tending towards infinity, and that λ0 ≥ 0. Recall that an = λn+1 − λn is the associated series whose partial sums form the sequence λ. Suppose
converges for all positive real numbers x. Then the Abelian mean Aλ is defined as
A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heat-kernel regularization.
Abelian means are regular, linear, and stable, but not always consistent between different choices of λ. However, some special cases are very important summation methods.
Read more about this topic: Divergent Series
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