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Lecture

Theorem of the Two Gendarmes: Convergence and Limits

In course

MATH-101(c): 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 theorem of the two gendarmes, which states that if two sequences converge to the same limit, then a third sequence formed by their sum also converges to that limit. It explores the convergence of sequences, properties of limits, and infinite behaviors such as polynomial, logarithmic, and exponential growth. The instructor demonstrates how to determine the convergence of sequences and discusses the relationship between exponential and power functions in terms of convergence speed.

Instructor

Sacha Friedli

Diploma of Physics, Université de Lausanne, 1999 Phd, Institut de Physique Théorique, EPFL, 1999-2004 Post-doc, IMPA (Instituto Nacional de Matemática Pura e Aplicada), Rio de Janeiro, Brasil, 2004-2006 Adjunct professor, Departamento de Matemática (Universidade Federal de Minas Gerais), 2006-2016 Collaborateur Scientifique, Cours de Mathématiques Spéciales, EPFL, Lausanne, 2016-

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.

Exponential growth

Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth).

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

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.

Sequence space

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.

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