# Definition:Abel Summation Method

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## Definition

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The series:

- $\ds \sum a_n$

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:

- $\ds \sum_{k \mathop = 0}^\infty a_k x^k$

is convergent and:

- $\ds \lim_{x \mathop \to 1^-} \sum_{k \mathop = 0}^\infty a_k x^k = S$

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

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:

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

## Source of Name

This entry was named for Niels Henrik Abel.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Abel summation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Abel summation** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**Abel summation**

- Abel summation method.
*Encyclopedia of Mathematics*. URL: https://www.encyclopediaofmath.org/index.php?title=Abel_summation_method&oldid=36182