Limit of a sequence

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Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
. 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
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
(i.e. an infinite set of points labelled using the natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s) in some sort of mathematical object (for example the real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s or a vector space

Vector space

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The limit of a sequence is one of the oldest concepts in mathematical analysis

Mathematical analysis

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s) in some sort of mathematical object (for example the real number

Real number

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
) which has a concept of nearness (such as "all points within a given distance of a fixed point"). 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.

Formal definition

For a sequence of real numbers

A real number L is said to be the limit of the sequence x_n, written

if and only if for every real number ε > 0, there exists a natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
N such that for every n > N we have |x_n−L| < ε.

For a sequence of points in a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
M with distance function d (such as a sequence of rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s, real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, points in a normed space, etc.):

An element is said to be the the limit of the sequence, written

if and only if for every real number ε > 0, there exists a natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
N such that for every n > N, we have d(x_n,L) < ε.

As a generalization of this, for a sequence of points in a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
T:

An element is said to be a limit of this sequence, written

if and only if for every neighborhood

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a Set containing the point where you can move that point some amount without leaving the set....
S of L there is a natural number N such that for all

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent (see also oscillation

Oscillation (mathematics)

In mathematics, oscillation is the behaviour of a sequence of real numbers or a real-valued function , which does not convergence, but also does not divergent series to +∞ or -∞; that is, oscillation is the failure to have a Limit , and is also a quantitative measure for that....
).

A null sequence is a sequence that converges to 0.

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

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 progresses....
).

A sequence of real numbers may tend to or , compare . Even though this can be written in the form

and

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

Real projective line

In real analysis, the real projective line , is the set , also denoted by and by .The symbol represents the point at infinity, an idealized point that bridges the two "ends" of the real line....
. 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

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
, for example the (extended) real line, the complex plane

Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
, their subsets (R
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
, Q
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
, Z
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
...) and Cartesian products (Rⁿ...).

The limit of a sequence of points in a topological space T is a special case of the : the domain is in the space with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +8, which in this space is a limit point

Limit point

In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x itself....
of .

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
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
.
If a is a real number with absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
|a| < 1, then the sequence aⁿ has limit 0. If 0 < a, then the sequence a^1/n has limit 1.

Also:

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

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function near a particular independent variable....
of f(x) at infinity.

A function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
f, defined on a first-countable space

First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space, X, is said to be first-countable if each point has a countable neighbourhood system ....
, is continuous

Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
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.e. implies 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

A subsequence
Subsequence

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements....
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. 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

Metric space

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 progresses....
and hence bounded

Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
. 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.

A sequence of real numbers is convergent if and only if its limit superior and limit inferior

Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. The limit inferior and limit superior of a function can be thought of in a similar fashion The limit inferior and limit superior of a set are the infimum and supremum of the set's limit points respectively....
coincide and are both finite.

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

then

and (if L₂ and y_n is non-zero)

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

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +8 and −8 ....
).

History

The Greek philosopher Zeno of Elea

Zeno of Elea

Zeno of Velia was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic....
is famous for formulating paradoxes that involve limiting processes

Zeno's paradoxes

Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion....
.

Leucippus

Leucippus

Leucippus or Leukippos was the first to develop the theory of atomism ? the idea that everything is composed entirely of various imperishable, indivisible elements called atoms ? which was elaborated in far greater detail by his pupil and successor, Democritus....
, Democritus

Democritus

Democritus was an Ancient Greek philosopher born in Abdera in the north of Greece. He was the most prolific, and ultimately the most influential, of the pre-Socratic philosophers; his atomic theory may be regarded as the culmination of early Greek thought....
, Antiphon

Antiphon (person)

Antiphon the Sophist lived in Athens probably in the last two decades of the 5th century BC. There is an ongoing controversy over whether he is one and the same with Antiphon of the Athenian deme Rhamnus in Attica, Greece , the earliest of the ten Attic orators....
, Eudoxus

Eudoxus

Eudoxus was the name of two ancient Greece:* Eudoxus of Cnidus , Greek astronomer and mathematician.* Eudoxus of Cyzicus , Greek navigator....
and Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
developed the method of exhaustion

Method of exhaustion

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas Convergence to the area of the containing shape....
, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series

Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive term . For example, the seriesis geometric, because each term is equal to half of the previous term....
.

Newton

Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
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). 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, mathematician

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s like Euler

Leonhard Euler

Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
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. At the end of the century, Lagrange

Joseph Louis Lagrange

Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia was an Italy mathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of mathematical analysis, to number theory, and to classical mechanics and celestial mechanics....
in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
in his etude of hypergeometric series

Hypergeometric series

In mathematics, a hypergeometric series, in the most general sense, is a power series in which the ratio of successive coefficients indexed by n is a rational function of n....
(1813) for the first time rigorously investigated under which conditions a series converged to a limit.

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).

Limit of a sequence

Formal definition

Comments

Examples

Properties

History

See also

External links