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In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ${\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}}$ is said to converge absolutely if ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L}$ for some real number ${\displaystyle \textstyle L.}$ Similarly, an improper integral of a function, ${\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx,}$ is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ${\displaystyle \textstyle \int _{0}^{\infty }|f(x)|dx=L.}$ A convergent series that is not absolutely convergent is called conditionally convergent.

Absolute convergence is important for the study of infinite series because its definition is strong enough to guarantee that a series will have some properties of finite sums that not all convergent series possess—absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series ${\textstyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots }$ converges to ${\displaystyle \ln 2,}$ while its rearrangement ${\textstyle 1+{\frac {1}{3}}-{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{4}}+\cdots }$ (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to ${\textstyle {\frac {3}{2}}\ln 2.}$

Background

In finite sums, the order in which terms are added is both associative and commutative, meaning that grouping and rearranging do not matter. (1 + 2) + 3 is the same as 1 + (2 + 3), and both are the same as (3 + 2) + 1. However, this is not true when adding infinitely many numbers, and wrongly assuming that it is true can lead to apparent paradoxes. One classic example is the alternating sum

${\displaystyle S=1-1+1-1+1-1...}$

whose terms alternate between +1 and −1. What is the value of S? One way to evaluate S is to group the first and second term, the third and fourth, and so on:

${\displaystyle S_{1}=(1-1)+(1-1)+(1-1)....=0+0+0...=0}$

But another way to evaluate S is to leave the first term alone and group the second and third term, then the fourth and fifth term, and so on:

${\displaystyle S_{2}=1+(-1+1)+(-1+1)+(-1+1)....=1+0+0+0...=1}$

This leads to an apparent paradox: does ${\displaystyle S=0}$ or ${\displaystyle S=1}$?

The answer is that because S is not absolutely convergent, grouping or rearranging its terms changes the value of the sum. This means ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ are not equal. In fact, the series ${\displaystyle 1-1+1-1+...}$ does not converge, so S does not have a value to find in the first place. A series that is absolutely convergent does not have this problem: grouping or rearranging its terms does not change the value of the sum.

Definition for real and complex numbers

A sum of real numbers or complex numbers ${\textstyle \sum _{n=0}^{\infty }a_{n}}$ is absolutely convergent if the sum of the absolute values of the terms ${\textstyle \sum _{n=0}^{\infty }|a_{n}|}$ converges.

Sums of more general elements

The same definition can be used for series ${\textstyle \sum _{n=0}^{\infty }a_{n}}$ whose terms ${\displaystyle a_{n}}$ are not numbers but rather elements of an arbitrary abelian topological group. In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function ${\textstyle \|\cdot \|:G\to \mathbb {R} _{+}}$ on an abelian group ${\displaystyle G}$ (written additively, with identity element 0) such that:

1. The norm of the identity element of ${\displaystyle G}$ is zero: ${\displaystyle \|0\|=0.}$
2. For every ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Absolute_convergence
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