Abel summation method

One of the methods for the summation of series of numbers. The series $$ \sum_{k=0}^\infty a_k $$ can be summed by the Abel method ($A$-method) to the number $A$ if, for any real $x$, $0<x<1$, the series $$ \sum_{k=0}^\infty a_k x^k $$ is convergent and $$ \lim_{x\rightarrow 1-0} \sum_{k=0}^\infty a_k x^k = S. $$ This summation method can already be found in the works of L. Euler and G. Leibniz. The name "Abel summation method" originates from the Abel theorem on the continuity of power series. The Abel summation method belongs to the class of totally regular summation methods and is more powerful than the entire set of Cesàro summation methods. The Abel summation method is used in conjunction with Tauberian theorems to demonstrate the convergence of a series.

A closely related summation method is the $A^*$-method. Let $z$ be a complex number, $\left|z\right|<1$; the series $$ \sum_{k=0}^\infty a_k $$ is summed by the $A^*$-method to the number $S$ if $$ \lim \sum_{k=0}^\infty a_k z^k = S, $$ where $z\rightarrow 1$ along any path not tangent to the unit circle.

References

[Ba]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
[Ha]	G.H. Hardy, "Divergent series", Clarendon Press (1949)