Abel sum

plural Abel sums

1. mathematical analysisGiven 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).

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

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

   2 more examples

   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