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