Meaning of Abel sum | Babel Free

Definitions

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).

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.”

“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.”

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

CEFR level