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Limit of a sequence

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Value to which tends an infinite sequence

For the general mathematical concept, seeLimit (mathematics).

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diagram of a hexagon and pentagon circumscribed outside a circle — The sequence given by the perimeters of regularn-sidedpolygons thatcircumscribe theunit circle has a limit equal to the perimeter of the circle, i.e. $2\pi$ . The corresponding sequence for inscribed polygons has the same limit.

{\displaystyle 2\pi } — The sequence given by the perimeters of regularn-sidedpolygons thatcircumscribe theunit circle has a limit equal to the perimeter of the circle, i.e. $2\pi$ . The corresponding sequence for inscribed polygons has the same limit.

$n {\displaystyle n}$	$n\times \sin \left({\tfrac {1}{n}}\right)$
1	0.841471
2	0.958851
...
10	0.998334
...
100	0.999983

As the positiveinteger ${\textstyle n}$ becomes larger and larger, the value ${\textstyle n\times \sin \left({\tfrac {1}{n}}\right)}$ becomes arbitrarily close to ${\textstyle 1}$ . We say that "the limit of the sequence ${\textstyle n\times \sin \left({\tfrac {1}{n}}\right)}$ equals ${\textstyle 1}$ ."

Inmathematics, thelimit of a sequence is the value that the terms of asequence "tend to", and is often denoted using the $\lim$ symbol (e.g., $\lim _{n\to \infty }a_{n}$ ).^[1] If such a limit exists and is finite, the sequence is calledconvergent.^[2] A sequence that does not converge is said to bedivergent.^[3] The limit of a sequence is said to be the fundamental notion on which the whole ofmathematical analysis ultimately rests.^[1]

Limits can be defined in anymetric ortopological space, but are usually first encountered in thereal numbers.

History

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The Greek philosopherZeno of Elea is famous for formulatingparadoxes that involve limiting processes.

Leucippus,Democritus,Antiphon,Eudoxus, andArchimedes developed themethod of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called ageometric series.

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of ageometric series in his workOpus Geometricum (1647): "Theterminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."^[4]

Pietro Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions inGeometriae speciosae elementa (1659). He used the termquasi-infinite forunbounded andquasi-null forvanishing.

Newton dealt with series in his works onAnalysis 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) andTractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to hisOptiks). In the latter work, Newton considers the binomial expansion of ${\textstyle (x+o)^{n}}$ , which he then linearizes bytaking the limit as ${\textstyle o}$ tends to ${\textstyle 0}$ .

In the 18th century,mathematicians such asEuler succeeded in summing somedivergent 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 hisThéorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus.Gauss in his study ofhypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.

The modern definition of a limit (for any ${\textstyle \varepsilon }$ there exists an index ${\textstyle N}$ so that ...) was given byBernard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and byKarl Weierstrass in the 1870s.

Real numbers

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The plot of a convergent sequence {*a_n*} is shown in blue. Here, one can see that the sequence is converging to the limit 0 asn increases.

In thereal numbers, a number $L {\displaystyle L}$ is thelimit of thesequence $(x_{n})$ , if the numbers in the sequence become closer and closer to $L {\displaystyle L}$ , and not to any other number.

Examples

[edit]

Definition

[edit]

We call $x {\displaystyle x}$ thelimit of thesequence $(x_{n})$ , which is written

x_{n}\to x

, or

\lim _{n\to \infty }x_{n}=x

if the following condition holds:

For eachreal number

\varepsilon >0

, there exists anatural number

N {\displaystyle N}

such that, for every natural number

n\geq N

, we have

|x_{n}-x|<\varepsilon

.^[6]

In other words, for every measure of closeness $\varepsilon$ , the sequence's terms are eventually that close to the limit. The sequence $(x_{n})$ is said toconverge to ortend to the limit $x {\displaystyle x}$ .

Symbolically, this is:

\forall \varepsilon >0\left(\exists N\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies |x_{n}-x|<\varepsilon \right)\right)\right)

If a sequence $(x_{n})$ converges to some limit $x {\displaystyle x}$ , then it isconvergent and $x {\displaystyle x}$ is the only limit; otherwise $(x_{n})$ isdivergent. A sequence that has zero as its limit is sometimes called anull sequence.

Illustration

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$Example of a sequence which converges to the limit a {\displaystyle a} .$
Example of a sequence which converges to the limit $a {\displaystyle a}$
Regardless which $\varepsilon >0$ we have, there is an index $N_{0}$ , so that the sequence lies afterwards completely in the epsilon tube $(a-\varepsilon ,a+\varepsilon )$ .
There is also for a smaller $\varepsilon _{1}>0$ an index $N_{1}$ , so that the sequence is afterwards inside the epsilon tube $(a-\varepsilon _{1},a+\varepsilon _{1})$ .
For each $\varepsilon >0$ there are only finitely many sequence members outside the epsilon tube.

Properties

[edit]

Some other important properties of limits of real sequences include the following:

When it exists, the limit of a sequence is unique.^[5]
Limits of sequences behave well with respect to the usualarithmetic operations. If $\lim _{n\to \infty }a_{n}$ and $\lim _{n\to \infty }b_{n}$ exists, then

\lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}

^[5]

\lim _{n\to \infty }ca_{n}=c\cdot \lim _{n\to \infty }a_{n}

^[5]

\lim _{n\to \infty }(a_{n}\cdot b_{n})=\left(\lim _{n\to \infty }a_{n}\right)\cdot \left(\lim _{n\to \infty }b_{n}\right)

^[5]

\lim _{n\to \infty }\left({\frac {a_{n}}{b_{n}}}\right)={\frac {\lim \limits _{n\to \infty }a_{n}}{\lim \limits _{n\to \infty }b_{n}}}

provided

\lim _{n\to \infty }b_{n}\neq 0

^[5]

\lim _{n\to \infty }a_{n}^{p}=\left(\lim _{n\to \infty }a_{n}\right)^{p}

For anycontinuous function ${\textstyle f}$ , if $\lim _{n\to \infty }x_{n}$ exists, then $\lim _{n\to \infty }f\left(x_{n}\right)$ exists too. In fact, any real-valuedfunction ${\textstyle f}$ is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).

If $a_{n}\leq b_{n}$ for all $n {\displaystyle n}$ greater than some $N {\displaystyle N}$ , then $\lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}$ .
(Squeeze theorem) If $a_{n}\leq c_{n}\leq b_{n}$ for all $n {\displaystyle n}$ greater than some $N {\displaystyle N}$ , and $\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L$ , then $\lim _{n\to \infty }c_{n}=L$ .
(Monotone convergence theorem) If $a_{n}$ isbounded andmonotonic for all $n {\displaystyle n}$ greater than some $N {\displaystyle N}$ , then it is convergent.
A sequence is convergent if and only if every subsequence is convergent.
If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.

These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example, once it is proven that $1/n\to 0$ , it becomes easy to show—using the properties above—that ${\frac {a}{b+{\frac {c}{n}}}}\to {\frac {a}{b}}$ (assuming that $b\neq 0$ ).

Infinite limits

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A sequence $(x_{n})$ is said totend to infinity, written

x_{n}\to \infty

, or

\lim _{n\to \infty }x_{n}=\infty

if the following holds:

For every real number

K {\displaystyle K}

, there is a natural number

N {\displaystyle N}

such that for every natural number

n\geq N

, we have

x_{n}>K

; that is, the sequence terms are eventually larger than any fixed

K {\displaystyle K}

Symbolically, this is:

\forall K\in \mathbb {R} \left(\exists N\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies x_{n}>K\right)\right)\right)

Similarly, we say a sequencetends to minus infinity, written

x_{n}\to -\infty

, or

\lim _{n\to \infty }x_{n}=-\infty

if the following holds:

For every real number

K {\displaystyle K}

, there is a natural number

N {\displaystyle N}

such that for every natural number

n\geq N

, we have

x_{n}<K

; that is, the sequence terms are eventually smaller than any fixed

K {\displaystyle K}

Symbolically, this is:

\forall K\in \mathbb {R} \left(\exists N\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies x_{n}<K\right)\right)\right)

If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence $x_{n}=(-1)^{n}$ provides one such example.

Metric spaces

[edit]

Definition

[edit]

A point $x {\displaystyle x}$ of themetric space $(X,d)$ is thelimit of thesequence $(x_{n})$ if:

For eachreal number

\varepsilon >0

, there is anatural number

N {\displaystyle N}

such that, for every natural number

n\geq N

, we have

d(x_{n},x)<\varepsilon

Symbolically, this is:

\forall \varepsilon >0\left(\exists N\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies d(x_{n},x)<\varepsilon \right)\right)\right)

This coincides with the definition given for real numbers when $X=\mathbb {R}$ and $d(x,y)=|x-y|$ .

Properties

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When it exists, the limit of a sequence is unique, as distinct points are separated by some positive distance, so for $\varepsilon$ less than half this distance, sequence terms cannot be within a distance $\varepsilon$ of both points.

For anycontinuous functionf, if $\lim _{n\to \infty }x_{n}$ exists, then $\lim _{n\to \infty }f(x_{n})=f\left(\lim _{n\to \infty }x_{n}\right)$ . In fact, afunctionf is continuous if and only if it preserves the limits of sequences.

Cauchy sequences

[edit]

Main article:Cauchy sequence

The plot of a Cauchy sequence (*x_n*), shown in blue, as $x_{n}$ versusn. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together asn increases. In thereal numbers every Cauchy sequence converges to some limit.

{\displaystyle x_{n}} — The plot of a Cauchy sequence (*x_n*), shown in blue, as $x_{n}$ versusn. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together asn increases. In thereal numbers every Cauchy sequence converges to some limit.

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences inmetric spaces, and, in particular, inreal analysis. One particularly important result in real analysis is theCauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in othercomplete metric spaces.

Topological spaces

[edit]

Definition

[edit]

A point $x\in X$ of the topological space $(X,\tau )$ is alimit orlimit point^[7]^[8] of thesequence $\left(x_{n}\right)_{n\in \mathbb {N} }$ if:

For everyneighbourhood

U {\displaystyle U}

x {\displaystyle x}

, there exists some

N\in \mathbb {N}

such that for every

n\geq N

, we have

x_{n}\in U

.^[9]

This coincides with the definition given for metric spaces, if $(X,d)$ is a metric space and $\tau$ is the topology generated by $d {\displaystyle d}$ .

A limit of a sequence of points $\left(x_{n}\right)_{n\in \mathbb {N} }$ in a topological space $T {\displaystyle T}$ is a special case of alimit of a function: thedomain is $\mathbb {N}$ in the space $\mathbb {N} \cup \lbrace +\infty \rbrace$ , with theinduced topology of theaffinely extended real number system, therange is $T {\displaystyle T}$ , and the function argument $n {\displaystyle n}$ tends to $+\infty$ , which in this space is alimit point of $\mathbb {N}$ .

Properties

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In aHausdorff space, limits of sequences are unique whenever they exist. This need not be the case in non-Hausdorff spaces; in particular, if two points $x {\displaystyle x}$ and $y {\displaystyle y}$ aretopologically indistinguishable, then any sequence that converges to $x {\displaystyle x}$ must converge to $y {\displaystyle y}$ and vice versa.

Hyperreal numbers

[edit]

The definition of the limit using thehyperreal numbers formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence $(x_{n})$ tends toL if for every infinitehypernatural ${\textstyle H}$ , the term $x_{H}$ is infinitely close to ${\textstyle L}$ (i.e., the difference $x_{H}-L$ isinfinitesimal). Equivalently,L is thestandard part of $x_{H}$ :

L={\rm {st}}(x_{H})

Thus, the limit can be defined by the formula

\lim _{n\to \infty }x_{n}={\rm {st}}(x_{H})

where the limit exists if and only if the righthand side is independent of the choice of an infinite ${\textstyle H}$ .

Sequence of more than one index

[edit]

Sometimes one may also consider a sequence with more than one index, for example, a double sequence $(x_{n,m})$ . This sequence has a limit $L {\displaystyle L}$ if it becomes closer and closer to $L {\displaystyle L}$ when bothn andm becomes very large.

Example

[edit]

If $x_{n,m}=c$ for constant ${\textstyle c}$ , then $x_{n,m}\to c$ .
If $x_{n,m}={\frac {1}{n+m}}$ , then $x_{n,m}\to 0$ .
If $x_{n,m}={\frac {n}{n+m}}$ , then the limit does not exist. Depending on the relative "growing speed" of ${\textstyle n}$ and ${\textstyle m}$ , this sequence can get closer to any value between ${\textstyle 0}$ and ${\textstyle 1}$ .

Definition

[edit]

We call $x {\displaystyle x}$ thedouble limit of thesequence $(x_{n,m})$ , written

x_{n,m}\to x

, or

\lim _{\begin{smallmatrix}n\to \infty \\m\to \infty \end{smallmatrix}}x_{n,m}=x

if the following condition holds:

For eachreal number

\varepsilon >0

, there exists anatural number

N {\displaystyle N}

such that, for every pair of natural numbers

n,m\geq N

, we have

|x_{n,m}-x|<\varepsilon

.^[10]

In other words, for every measure of closeness $\varepsilon$ , the sequence's terms are eventually that close to the limit. The sequence $(x_{n,m})$ is said toconverge to ortend to the limit $x {\displaystyle x}$ .

Symbolically, this is:

\forall \varepsilon >0\left(\exists N\in \mathbb {N} \left(\forall n,m\in \mathbb {N} \left(n,m\geq N\implies |x_{n,m}-x|<\varepsilon \right)\right)\right)

The double limit is different from taking limit inn first, and then inm. The latter is known asiterated limit. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.

Infinite limits

[edit]

A sequence $(x_{n,m})$ is said totend to infinity, written

x_{n,m}\to \infty

, or

\lim _{\begin{smallmatrix}n\to \infty \\m\to \infty \end{smallmatrix}}x_{n,m}=\infty

if the following holds:

For every real number

K {\displaystyle K}

, there is a natural number

N {\displaystyle N}

such that for every pair of natural numbers

n,m\geq N

, we have

x_{n,m}>K

; that is, the sequence terms are eventually larger than any fixed

K {\displaystyle K}

Symbolically, this is:

\forall K\in \mathbb {R} \left(\exists N\in \mathbb {N} \left(\forall n,m\in \mathbb {N} \left(n,m\geq N\implies x_{n,m}>K\right)\right)\right)

Similarly, a sequence $(x_{n,m})$ tends to minus infinity, written

x_{n,m}\to -\infty

, or

\lim _{\begin{smallmatrix}n\to \infty \\m\to \infty \end{smallmatrix}}x_{n,m}=-\infty

if the following holds:

For every real number

K {\displaystyle K}

, there is a natural number

N {\displaystyle N}

such that for every pair of natural numbers

n,m\geq N

, we have

x_{n,m}<K

; that is, the sequence terms are eventually smaller than any fixed

K {\displaystyle K}

Symbolically, this is:

\forall K\in \mathbb {R} \left(\exists N\in \mathbb {N} \left(\forall n,m\in \mathbb {N} \left(n,m\geq N\implies x_{n,m}<K\right)\right)\right)

If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence $x_{n,m}=(-1)^{n+m}$ provides one such example.

Pointwise limits and uniform limits

[edit]

For a double sequence $(x_{n,m})$ , we may take limit in one of the indices, say, $n\to \infty$ , to obtain a single sequence $(y_{m})$ . In fact, there are two possible meanings when taking this limit. The first one is calledpointwise limit, denoted

x_{n,m}\to y_{m}\quad {\text{pointwise}}

, or

\lim _{n\to \infty }x_{n,m}=y_{m}\quad {\text{pointwise}}

which means:

For eachreal number

\varepsilon >0

and each fixednatural number

m {\displaystyle m}

, there exists a natural number

N(\varepsilon ,m)>0

such that, for every natural number

n\geq N

, we have

|x_{n,m}-y_{m}|<\varepsilon

.^[11]

Symbolically, this is:

\forall \varepsilon >0\left(\forall m\in \mathbb {N} \left(\exists N\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies |x_{n,m}-y_{m}|<\varepsilon \right)\right)\right)\right)

When such a limit exists, we say the sequence $(x_{n,m})$ converges pointwise to $(y_{m})$ .

The second one is calleduniform limit, denoted

x_{n,m}\to y_{m}\quad {\text{uniformly}}

\lim _{n\to \infty }x_{n,m}=y_{m}\quad {\text{uniformly}}

x_{n,m}\rightrightarrows y_{m}

, or

{\underset {n\to \infty }{\mathrm {unif} \lim }}\;x_{n,m}=y_{m}

which means:

For eachreal number

\varepsilon >0

, there exists a natural number

N(\varepsilon )>0

such that, for everynatural number

m {\displaystyle m}

and for every natural number

n\geq N

, we have

|x_{n,m}-y_{m}|<\varepsilon

.^[11]

Symbolically, this is:

\forall \varepsilon >0\left(\exists N\in \mathbb {N} \left(\forall m\in \mathbb {N} \left(\forall n\in \mathbb {N} \left(n\geq N\implies |x_{n,m}-y_{m}|<\varepsilon \right)\right)\right)\right)

In this definition, the choice of $N {\displaystyle N}$ is independent of $m {\displaystyle m}$ . In other words, the choice of $N {\displaystyle N}$ isuniformly applicable to all natural numbers $m {\displaystyle m}$ . Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit:

x_{n,m}\to y_{m}

uniformly, then

x_{n,m}\to y_{m}

pointwise.

When such a limit exists, we say the sequence $(x_{n,m})$ converges uniformly to $(y_{m})$ .

Iterated limit

[edit]

For a double sequence $(x_{n,m})$ , we may take limit in one of the indices, say, $n\to \infty$ , to obtain a single sequence $(y_{m})$ , and then take limit in the other index, namely $m\to \infty$ , to get a number $y {\displaystyle y}$ . Symbolically,

\lim _{m\to \infty }\lim _{n\to \infty }x_{n,m}=\lim _{m\to \infty }y_{m}=y

This limit is known asiterated limit of the double sequence. The order of taking limits may affect the result, i.e.,

\lim _{m\to \infty }\lim _{n\to \infty }x_{n,m}\neq \lim _{n\to \infty }\lim _{m\to \infty }x_{n,m}

in general.

A sufficient condition of equality is given by theMoore-Osgood theorem, which requires the limit $\lim _{n\to \infty }x_{n,m}=y_{m}$ to be uniform in ${\textstyle m}$ .^[10]

Notes

[edit]

^^a ^bCourant (1961), p. 29.
^Weisstein, Eric W."Convergent Sequence".mathworld.wolfram.com. Retrieved2020-08-18.
^Courant (1961), p. 39.
^Van Looy, H. (1984). A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667). Historia Mathematica, 11(1), 57-75.
^^a ^b ^c ^d ^e ^f ^g"Limits of Sequences | Brilliant Math & Science Wiki".brilliant.org. Retrieved2020-08-18.
^Weisstein, Eric W."Limit".mathworld.wolfram.com. Retrieved2020-08-18.
^Dugundji 1966, pp. 209–210.
^Császár 1978, p. 61.
^Zeidler, Eberhard (1995).Applied functional analysis : main principles and their applications (1 ed.). New York: Springer-Verlag. p. 29.ISBN 978-0-387-94422-7.
^^a ^bZakon, Elias (2011). "Chapter 4. Function Limits and Continuity".Mathematical Anaylysis, Volume I. p. 223.ISBN 9781617386473.
^^a ^bHabil, Eissa (2005)."Double Sequences and Double Series". Retrieved2022-10-28.

Proofs

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^Proof: Choose $N=1$ . For every $n\geq N$ , $|x_{n}-c|=0<\varepsilon$
^Proof: Choose an integer $N>{\frac {1}{\varepsilon }}.$ For every $n\geq N$ , one has $|x_{n}-0|={\frac {1}{n}}\leq {\frac {1}{N}}<\varepsilon$ .

References

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Császár, Ákos (1978).General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd.ISBN 0-85274-275-4.OCLC 4146011.
Dugundji, James (1966).Topology. Boston: Allyn and Bacon.ISBN 978-0-697-06889-7.OCLC 395340485.
Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow.
Frank Morley andJames Harkness A treatise on the theory of functions (New York: Macmillan, 1893)

External links

[edit]

"Limit",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
A history of the calculus, including limits

Calculus

Precalculus

Limits

Differential calculus

Integral calculus

Vector calculus

Derivatives
Basic theorems

Multivariable calculus

Sequences and series

Special functions
and numbers

History of calculus

Lists

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