Definition:Abel Summation Method

Definition
The series:


 * $\displaystyle \sum a_n$

can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$ such that $0 < x < 1$, the series:
 * $\displaystyle \sum_{k \mathop = 0}^\infty a_k x^k$

is convergent and:
 * $\displaystyle \lim_{x \mathop \to 1^-} \sum_{k \mathop =0}^\infty a_k x^k = S$


 * $\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n e^{-n x} = \sum_{n \mathop = 0}^\infty a_n z^n$

where $z = \map \exp {−x}$.

Then the limit of $\map f x$ as $x$ approaches $0$ through positive reals is the limit of the power series for $\map f z$ as $z$ approaches $1$ from below through positive reals.

The Abel sum $\map A s$ is defined as:


 * $\displaystyle \map A s = \lim_{z \mathop \to 1^-} \sum_{n \mathop = 0}^\infty a_n z^n$