Limit of a sequence - Citizendia

The limit of a sequence is one of the oldest concepts in mathematical analysis. Analysis has its beginnings in the rigorous formulation of Calculus. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit.

Intuitively, suppose we have a sequence of points (i. In Mathematics, a sequence is an ordered list of objects (or events e. an infinite set of points labelled using the natural numbers) in some sort of mathematical object (for example the real numbers or a vector space) which has a concept of nearness (such as "all points within a given distance of a fixed point"). In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A point L is the limit of the sequence if for any prescribed nearness, all but a finite number of points in the sequence are that near to L. This may be visualised as a set of spheres of size decreasing to zero, all with the same centre L, and for any one of these spheres, only a finite number of points in the sequence being outside the sphere.

1 Formal definition
- 1.1 Comments
2 Examples
3 Properties
4 History
5 References
6 See also
7 External links

Formal definition

For a sequence of points $\{x_n|n\in \mathbb{N}\}\;$ in a metric space M with distance function d (such as a sequence of rational numbers, real numbers, complex numbers, points in a normed space, etc. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to ):

If $L\in M\;$ we say L is the limit of the sequence and write

$L = \lim_{n \to \infty} x_n$

$\Longleftrightarrow \forall \epsilon>0\;, \exists N \in \mathbb{N}: n>N \rightarrow d(x_n,L)<\epsilon.\;$

i. e. : if and only if for every real number $\epsilon>0\;$ , there exists a natural number N such that for every $n>N\;$ , we have $d(x_n,L)<\epsilon.\;$

As a generalization of this, for a sequence of points $\{x_n|n\in \mathbb{N}\}\;$ in a topological space T:

If $L\in T\;$ we say L is a limit of this sequence and write

$L = \lim_{n \to \infty} x_n$

$\Longleftrightarrow \forall U(L) \; \exists N \in \mathbb{N}: \forall n > N \; x_n \in U(L)$

i. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. e. : if and only if for every neighborhood S of L there is a natural number N such that $x_n\in S\;$ for all $n>N.\;$

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Otherwise, the sequence is divergent (see also oscillation). In Mathematics, oscillation is the behaviour of a Sequence of Real numbers or a real-valued function, which does not converge,

A null sequence is a sequence that converges to 0. *Broken Link* ^[1]

Comments

The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence). In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence

A sequence of real numbers may tend to $+\infty$ or $-\infty$ , compare infinite limits. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular Even though this can be written in the form

$\lim_{n \to \infty} x_n = \infty$ and $\lim_{n \to \infty} x_n = -\infty$

such a sequence is called divergent, unless we explicitly consider it a sequence in the affinely extended real number system or (in the first case only) the real projective line. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Real analysis, the real projective line (also called the one-point compactification of the Real line, or the projectively extended real numbers In the latter cases the sequence has a limit (in the space itself), so could be called convergent, but when using this term here, care should be taken that this does not cause confusion.

Also, a sequence may, in a general topological space, have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space, for example the (extended) real line, the complex plane, their subsets (R, Q, Z. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French . . ) and Cartesian products (Rⁿ. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural . . ).

The limit of a sequence of points $\{x_n|n\in \mathbb{N}\}\;$ in a topological space T is a special case of the limit of a function: the domain is $\mathbb{N}$ in the space $\mathbb{N} \cup \lbrace +\infty \rbrace$ with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +∞, which in this space is a limit point of $\mathbb{N}$ . In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"

Examples

The sequence 1, -1, 1, -1, 1, . . . is divergent.
The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, . . . converges with limit 1. This is an example of an infinite series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with
If a is a real number with absolute value |a| < 1, then the sequence aⁿ has limit 0. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. If 0 < a ≤ 1, then the sequence a^1/n has limit 1.

Also:

$\lim_{n\to\infty} \frac{1}{n^p} = 0 \hbox{ if } p > 0$

$\lim_{n\to\infty} a^n = 0 \hbox{ if } |a| < 1$
$\lim_{n\to\infty} n^{\frac{1}{n}} = 1$
$\lim_{n\to\infty} a^{\frac{1}{n}} = 1 \hbox{ if } a>0$

Properties

Consider the following function: f(x)=x_n if n-1<x≤n. Then the limit of the sequence of x_n is just the limit of f(x) at infinity. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular

A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(x_n)) converges to f(L) given that (x_n) converges to L, i. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function e.

$\lim_{n\to\infty}x_n=L$ implies $\lim_{n\to\infty}f(x_n)=f(L)$

Note that this equivalence does not hold in general for spaces which are not first-countable.

Compare the basic property (or definition):

f is continuous at x if and only if $\lim_{x\to L}f(x)=f(L)$

A subsequence of the sequence (x_n) is a sequence of the form (x_a(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. In Mathematics, a subsequence of some Sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.

Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined A bounded monotonic sequence of real numbers is necessarily convergent: this is sometimes called the fundamental theorem of analysis. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has

A sequence of real numbers is convergent if and only if its limit superior and limit inferior coincide and are both finite. In Mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup

The algebraic operations are everywhere continuous (except for division around zero divisor); thus, given

$\lim_{n \to \infty}x_n = L_1$ and $\lim_{n \to \infty}y_n = L_2$

then

$\lim_{n \to \infty}(x_n+y_n) = L_1 + L_2$

$\lim_{n \to \infty}(x_ny_n) = L_1L_2$

and (if L₂ and y_n is non-zero)

$\lim_{n \to \infty}(x_n/y_n) = L_1/L_2$

These rules are also valid for infinite limits using the rules

q + ∞ = ∞ for q ≠ -∞
q × ∞ = ∞ if q > 0
q × ∞ = -∞ if q < 0
q / ∞ = 0 if q ≠ ± ∞

(see extended real number line). In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced

History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Zeno of Elea (ˈziːnoʊ əv ˈɛliə Greek: Ζήνων ὁ Ἐλεάτης (ca

Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Leucippus or Leukippos ( Greek, first half of 5th century BC was among the earliest philosophers of Atomism, the idea that everything is composed entirely Democritus ( Greek:) was a pre-Socratic Greek Materialist Philosopher (born at Abdera in Thrace ca This article is about the musical term See Antiphon (person the orator of ancient Greece Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Archimedes succeeded in summing what is now called a geometric series.

Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In the latter work, Newton considers the binomial expansion of (x+o)ⁿ which he then linearizes by taking limits (letting o→0).

In the 18th century, mathematicians like Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. A mathematician is a person whose primary area of study and research is the field of Mathematics. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated under which conditions a series converged to a limit. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function

The modern definition of a limit (for any ε there exists an index N so that . . . ) was given independently by Bernhard Bolzano (Der binomische Lehrsatz, Prag 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821). Bernard (Bernhard Placidus Johann Nepomuk Bolzano ( &ndash December 18, 1848) was a Bohemian Mathematician, theologian,

References

^ http://people.reed.edu/~mayer/html2/node10.html

External links

In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, there are many senses in which a sequence or a series is said to be convergent

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