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Concept
Absolute convergence
Summary
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 is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while 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 converges to while its rearrangement (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to
In finite sums, the order in which terms are added is associative, meaning that the order does not matter. 1 + 2 + 3 is 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
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:
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:
This leads to an apparent paradox: does or ?
The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means and are not equal. In fact, the series 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: rearranging its terms does not change the value of the sum.
Official source
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