Abel sum

English

Etymology

After Norwegian mathematician Template:W (1802-1829).

Noun

1. Template:Lb Given a power series $f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$ that is convergent for real x in the open interval (0, 1), the value $\underset{x\to 1^{-}}{\lim }\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$, which is assigned to $f(1)=\sum _{n=0}^{\mathrm{\infty }}{a}_n$ by the Abel summation method (or A-method).
   • 1967, Jan Mikusiński, Operational Calculus, Template:W, page 102,

     The Abel sum of $\sum a_n$ is defined as the limit of the corresponding power series:

     $\underset{x\to 1-0}{\lim }\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$.

     The existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r.

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Derived terms

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Related terms

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See also

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Further reading

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