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