Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Divergent Sequences

Description

This lecture covers the concept of divergent sequences, which are sequences that do not converge. It explains the definition of divergent sequences and provides examples, such as an = (-1), to illustrate this concept. The lecture also discusses the notion of limits at infinity and how to determine if a sequence is divergent or tends to positive or negative infinity.

In MOOCs (2)

Analysis I

Analysis I (part 3) : Real number sequences I and II

Official source

Quiz

Related concepts (19)

Limit of a sequence

As the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Construction of the real numbers

In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them.

Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability.

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.

Related lectures (30)

Limit of a Sequence MOOC: Analysis I MOOC: Analysis I (part 3) : Real number sequences I and II

Explains the concept of the limit of a sequence and its importance in convergence.

Limits of Sequences

Explores the concept of limits of sequences and their convergence towards infinity or negative infinity.

Two Propositions MOOC: Analysis I MOOC: Analysis I (part 3) : Real number sequences I and II

Discusses two propositions about converging sequences of real numbers and explores their consequences.

Cauchy Sequences MOOC: Analysis I MOOC: Analysis I (part 3) : Real number sequences I and II

Explores Cauchy sequences in real numbers, convergence criteria, and limit definitions.

Continuity Definition: Epsilon and Delta MOOC: Analysis I MOOC: Analysis I (part 4) : Limit of a function, continuous functions