# Abel sum

## English

### Etymology

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

### Noun

Abel sum (plural Abel sums)

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