Limit of a sequence
Encyclopedia
The limit
of a sequence
is, intuitively, the unique number or point L (if it exists) such that the terms of the sequence become arbitrarily close to L for "large" values of n. If the limit exists, then we say that the sequence is convergent and that it converges to L.
Convergence of sequences is a fundamental notion in mathematical analysis
, which has been studied since ancient times.
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
).
A null sequence is a sequence that converges to 0.
. Equivalently, L is the standard part of xH
.
Thus, the limit can be defined by the formula
where the limit exists if and only if the righthand side is independent of the choice of an infinite H.
).
A sequence of real numbers may tend to
or 
, compare infinite limits. 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. 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.
The limit of a sequence of points
in a topological space T is a special case of the limit of a function: 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 +∞, which in this space is a limit point
of
.
Also:
of f( x) at infinity.
A function
f, defined on a first-countable space
, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) 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):
A subsequence
of the sequence (xn) is a sequence of the form (xa(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
is a Cauchy sequence
and hence bounded
. 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
coincide and are both finite.
The algebraic operations are continuous everywhere (except for division around zero divisor); thus, given
and 
then
and (if L2 and yn are non-zero)
These rules are also valid for infinite limits using the rules
(see extended real number line
).
is famous for formulating paradoxes that involve limiting processes
.
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. 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). 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
s 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. 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.
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, Prague 1816, little noticed at the time) and by Cauchy
in his Cours d'analyse (1821).
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
of a sequence
is, intuitively, the unique number or point L (if it exists) such that the terms of the sequence become arbitrarily close to L for "large" values of n. If the limit exists, then we say that the sequence is convergent and that it converges to L.Convergence of sequences is a fundamental notion in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, which has been studied since ancient times.
Formal definition
- For a sequence of real numbers
- A real number L is said to be the limit of the sequence xn, written
- if and only if for every real number ε > 0, there exists a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
N such that for every n > N we have | xn−L | < ε.
- As a generalization of this, for a sequence of points
in a topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
T:
- An element is said to be a limit of this sequence if and only if for every neighborhoodNeighbourhood (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.This concept is closely related to the...
S of L there is a natural number N such that for all . In this generality a sequence may admit more than one limit, but if T is a Hausdorff spaceHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
, then each sequence has at most one limit. When a unique limit L exists, it may be written
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 converge, but also does not diverge to +∞ or −∞; that is, oscillation is the failure to have a limit, and is also a quantitative measure for that.Oscillation is defined as the...
).
A null sequence is a sequence that converges to 0.
Hyperreal definition
A sequence xn tends to L if for every infinite hypernatural H, the term xH is infinitely close to L, i.e., the difference xH - L is infinitesimalInfinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
. Equivalently, L is the standard part of xH
.Thus, the limit can be defined by the formula
where the limit exists if and only if the righthand side is independent of the choice of an infinite H.
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 sequenceCauchy 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...
).
A sequence of real numbers may tend to
or , compare infinite limits. 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. 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.
The limit of a sequence of points
in a topological space T is a special case of the limit of a function: the domain is in the space with the induced topologyInduced topology
In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space.- Definition :Let X_0, X_1 be sets, f:X_0\to X_1....
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
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 in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
of
.Examples
- The sequence 1, -1, 1, -1, 1, ... is oscillatory.
- The seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
with partial sums 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 seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
. - If a is a real number with absolute valueAbsolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
| a | < 1, then the sequence an has limit 0. If 0 < a, then the sequence a1/n has limit 1.
Also:
Properties
Consider the following function: f( x ) = xn if n-1 < x ≤ n. Then the limit of the sequence of xn is just the limitLimit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
of f( x) at infinity.
A function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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 basis...
, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) 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 (xn) is a sequence of the form (xa(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
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...
is a Cauchy sequence
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...
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...
coincide and are both finite.
The algebraic operations are continuous everywhere (except for division around zero divisor); thus, given
and then
and (if L2 and yn are 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: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
).
History
The Greek philosopher Zeno of EleaZeno of Elea
Zeno of Elea 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. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
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 Greek philosopher 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...
.
Leucippus
Leucippus
Leucippus or Leukippos was one of the earliest Greeks to develop the theory of atomism — the idea that everything is composed entirely of various imperishable, indivisible elements called atoms — which was elaborated in greater detail by his pupil and successor, Democritus...
, Democritus
Democritus
Democritus was an Ancient Greek philosopher born in Abdera, Thrace, Greece. He was an influential pre-Socratic philosopher and pupil of Leucippus, who formulated an atomic theory for the cosmos....
, 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 , the earliest of the ten Attic orators...
, Eudoxus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...
and Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, 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. Among his advances in physics are the foundations of hydrostatics, statics and an...
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 converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...
, 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.
Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
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 is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s like Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
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 a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical 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 German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
in his etude of hypergeometric series
Hypergeometric series
In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...
(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, Prague 1816, little noticed at the time) and by Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
in his Cours d'analyse (1821).
See also
- Limit of a functionLimit of a functionIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
- Limit of a net - a netNet (mathematics)In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
is a topological generalization of a sequence - Modes of convergenceModes of convergenceIn mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes of convergence in the settings where they are defined...