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Abel sum

Abel sum noun: given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method).

Noun

Abel sum (plural Abel sums)

  1. Given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method).
    • lim _(x→1-0)∑ₙ₌₀ ᪲a_nxⁿ.
      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.1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102, The Abel sum of ∑a_n is defined as the limit of the corresponding power series
    • Jacobi in his Vorlesungen über Dynamik [1884] had used Abel sums to separate variables in the Hamilton-Jacobi equation in connection with the geodesic flow on the surface of a 3-dimensional ellipsoid, etc.2005, Bulletin of the American Mathematical Society, page 81

Forms of Abel sum

pluralAbel sums

Origin of Abel sum

After Norwegian mathematician Niels Henrik Abel (1802-1829).

Words formed from Abel sum

  • Abel summable
  • Abel summation

Questions about Abel sum

What does Abel sum mean?
Abel sum means given a power series f(x)=∑ₙ₌₀ ᪲a_nxⁿ that is convergent for real x in the open interval (0, 1), the value lim _(x→1⁻)∑ₙ₌₀ ᪲a_nxⁿ, which is assigned to f(1)=∑ₙ₌₀ ᪲a_n by the Abel summation method (or A-method).
What part of speech is Abel sum?
Abel sum is a noun.
What is the plural of Abel sum?
The plural of Abel sum is Abel sums.

Words near Abel sum in the dictionary

  1. Abeita
  2. Abeitas
  3. Abel
  4. Abel polynomial
  5. Abel polynomials
  6. Abel sum
  7. Abel sums
  8. Abel test
  9. Abel tests
  10. Abel's formula
  11. Abel's identity

Source and license

Adapted from Wiktionary: Abel sum (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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