Lecture

Convergence of Series: Solutions and Tests

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In course

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. ...

Description

This lecture covers the convergence of series, providing solutions to problems and discussing various convergence tests such as the Cauchy test and comparison test. The instructor explains how to determine convergence based on odd and even terms.

Instructor

Thomas Mountford

Teaching & PhD Past EPFL PhD Students Damien Gauthier, Samir Brahim Belhaouari, Driss Baraka, Daniel Rodrigues Valesin, Samuel Schöpfer, Kamran Kalbasi, Jacques Saliba, Hao Xue, Keming Chen Past EPFL PhD Students as codirector Guillaume Charles Woessner Courses Analysis I MATH-101(d) Study the fundamental concepts of analysis, and the differential and integral calculus of real-valued functions of one real variable. Markov chains MATH-332 The course follows the text of Norris

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Root test

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series. The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test.

Direct comparison test

In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.

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.

Limit comparison test

In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Suppose that we have two series and with for all . Then if with , then either both series converge or both series diverge. Because we know that for every there is a positive integer such that for all we have that , or equivalently As we can choose to be sufficiently small such that is positive. So and by the direct comparison test, if converges then so does .

Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Consider an integer N and a function f defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series converges to a real number if and only if the improper integral is finite. In particular, if the integral diverges, then the series diverges as well.

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