Lecture

Convergence of Series

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In course

MATH-101(g): Analysis I

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

Description

This lecture covers the criteria for the convergence of numerical series, including the definition of convergent series, partial sums, and the sum of series. It also discusses absolute convergence, infinite sums, and the convergence criteria of series. The lecture explores the Cauchy criterion, Leibniz criterion, and alternating series, providing examples and demonstrations to illustrate the concepts.

Instructors (2)

Annalisa Buffa

Teaching & PhD PhD Students Gonzalo Bonilla Moreno, Philipp Christoph Weder, Yannis Dirk Voet, Fabio Marcinno, Ferhat Sindy, Francesco Sala, Markus Renoldner, Mohamed Ben Abdelouahab, Davor Dobrota Past EPFL PhD Students Luca Coradello, Ondine Chanon, Riccardo Puppi Courses Numerical approximation of PDEs MATH-451 The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in on

Lénaïc Chizat

Teaching & PhD PhD Students Javier Esteban Maass Martinez Past EPFL PhD Students Guillaume Yitian Wang Courses Computational optimal transport MATH-528 Computational aspects of measure transportation, from classical optimal transport to modern denoising diffusion models. Topics in machine learning MATH-520 Mathematical analysis of modern supervised machine learning techniques, with an emphasis on the mathematics of artificial neural networks.

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Alternating series

In mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not.

Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

Alternating series test

In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. A series of the form where either all an are positive or all an are negative, is called an alternating series.

Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series . If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.

Radius of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

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Convergence and Divergence of Series

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Convergence of Series

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Telescopic Series: Convergence and Divergence

Covers telescopic series, partial sums, convergence, and divergence using various examples and proofs.

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Related concepts (32)

Alternating series

Series (mathematics)

Alternating series test

Convergence tests

Radius of convergence