Abel sum

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

Abel sum (plural Abel sums)

1. (mathematical analysis) Given a power series ${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}}$ that is convergent for real x in the open interval (0, 1), the value ${\displaystyle \lim _{x\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}x^{n}}$, which is assigned to ${\displaystyle f(1)=\sum _{n=0}^{\infty }a_{n}}$ by the Abel summation method (or A-method).
   • 1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102,

     The Abel sum of ${\displaystyle \textstyle \sum a_{n}}$ is defined as the limit of the corresponding power series:

     ${\displaystyle \lim _{x\rightarrow 1-0}\sum _{n=0}^{\infty }a_{n}x^{n}}$.

     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.

   • 2005, Bulletin of the American Mathematical Society, page 81:

     Jacobi in his Vorlesungen über Dynamik 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.

   • 2012, Peter L. Duren, Invitation to Classical Analysis, American Mathematical Society, page 180:

     Also, Abel's theorem guarantees that the Abel sum of a convergent series exists and is equal to the ordinary sum.

• Abel summable
• Abel summation

• Abel mean
• Abel summation method

• summability method
• summation method

• Divergent series on Wikipedia.Wikipedia
• Abel's theorem on Wikipedia.Wikipedia
• Abel's summation formula on Wikipedia.Wikipedia

• bemauls, malesub