Lecture

Convergence Criteria

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers the convergence criteria for monotonic and bounded sequences of real numbers, demonstrating that a sequence is convergent if it is either increasing and bounded above or decreasing and bounded below.

In MOOCs (2)

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

A sequence of real numbers is a function f:N→R . It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, i.e. a0=1,a1=12,a2=14,a3=18,.... . The central concept is that of the limit of a sequence: this is a real number to which, intuitively, the given sequence gets closer and closer. For example, the sequence an given above admits the number zero as its limit. We'll define the concept of limit in a rigorous way and develop methods for establishing the exi

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

Official source

Quiz

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (22)

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.

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.

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.

Monotone convergence theorem

In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

Related lectures (64)

Sequence Convergence: Definitions and Properties

Covers definitions and properties of sequence convergence, including limits, uniqueness, and the two gendarmes theorem.

Subsequences and Bolzano-Weierstrass Theorem

Covers the proof of the Squeeze Theorem, Quotient Criteria, and the Bolzano-Weierstrass Theorem.

Real Numbers Sequences

Explores real number sequences, convergence, limits, boundedness, and monotonicity.