absolute convergence of a numerical series

绝对收敛

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A real or complex numerical series sum a_n is absolutely convergent when the nonnegative series sum |a_n| has a finite sum.

实数或复数项级数若各项绝对值组成的非负级数之和有限，则原级数绝对收敛。

Example

∑n=1∞(−1)n/3n\sum_{n=1}^\infty(-1)^n/3^n converges absolutely because ∑3−n=1/2\sum 3^{-n}=1/2.

Usage and distinctions

Conditional convergence means convergence of the original series without convergence of its absolute-value series.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§1.6, equation (1.6.40) and definition after Proposition 1.6.13 · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：一、数项级数，条件收敛与绝对收敛性 · Checked 2026-10-14
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