# Definition:Abel Summation Method

Jump to navigation
Jump to search

## Definition

This page needs the help of a knowledgeable authority.In particular: It is difficult finding a concise and complete definition of exactly what the Abel Summation Method actually is. All and any advice as to how to implement this adequately is requested of anyone. This is what is said in the Spring encyclopedia on the page "Abel summation method":If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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:

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

This page needs the help of a knowledgeable authority.In particular: This is what we have on Wikipedia page Divergent series:If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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

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**

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