English Dictionary

Abel sum

名詞

(mathematical analysis) 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).

日本語の意味

Abel和（エイベルわ）：数学解析における和の定義の一種。具体的には、冪級数 f(x)=∑ₙ₌₀∞ aₙxⁿ が (0,1) の区間で収束するとき、x→1⁻ の極限 limₓ→1⁻∑ₙ₌₀∞ aₙ xⁿ により f(1) が定義される手法を指す。

For the power series f(x)=∑{n=0}∞ a_n xn that converges for 0<x<1, the Abel sum is defined as lim{x→1-} ∑_{n=0}∞ a_n xn and is assigned to f(1) by the Abel summation method.

冪級数 f(x)=∑_{n=0}^∞ a_n x^n が 0

意味（1）

(mathematical analysis) 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).

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.

Abel sums （ plural ）