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
Famous quotes containing the word means:
“I don’t think Dr. King helped racial harmony, I think he helped racial justice. What I profess to do is help the oppressed and if I cause a load of discomfort in the white community and the black community, that in my opinion means I’m being effective, because I’m not trying to make them comfortable. The job of an activist is to make people tense and cause social change.”
—Al, Reverend Sharpton (b. 1954)