**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

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

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (103)

Related people (20)

Related units (2)

Related concepts (18)

Related courses (32)

Dans ce cours, nous étudierons les notions fondamentales de l'analyse réelle, ainsi que le calcul différentiel et intégral pour les fonctions réelles d'une variable réelle.

Nous étudions les concepts fondamentaux de l'analyse, le calcul différentiel et intégral de fonctions réelles d'une variable.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Related lectures (298)

Related MOOCs (13)

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.

Analyse I

Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond

Analyse I (partie 1) : Prélude, notions de base, les nombres réels

Concepts de base de l'analyse réelle et introduction aux nombres réels.

Analyse I (partie 2) : Introduction aux nombres complexes

Introduction aux nombres complexes

Covers absolute convergence, reordering series, and includes a quiz series.

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Covers the convergence criteria for sequences, including operations on limits and sequences defined by recurrence.

In this work we consider solutions to stochastic partial differential equations with transport noise, which are known to converge, in a suitable scaling limit, to solution of the corresponding deterministic PDE with an additional viscosity term. Large devi ...

Giancarlo Ferrari Trecate, Maryam Kamgarpour, Yuning Jiang, Baiwei Guo

We address black-box convex optimization problems, where the objective and constraint functions are not explicitly known but can be sampled within the feasible set. The challenge is thus to generate a sequence of feasible points converging towards an optim ...

2023We give an extension of Le's stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments o ...