Limit of a sequence

350px|right|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|The sequence given by the perimeters of regular n-sided [[polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. <math>2\pi</math>. The corresponding sequence for inscribed polygons has the same limit.]]

{| class="wikitable" style="width:100%;"

!<math>n</math>!!<math>n\times \sin\left(\tfrac1{n}\right)</math>

|1||0.841471

|2||0.958851

|colspan="2"|...

|10||0.998334

|colspan="2"|...

|100||0.999983

</div>

As the positive integer <math display="inline">n</math> becomes larger and larger, the value <math display="inline">n\times \sin\left(\tfrac1{n}\right)</math> becomes arbitrarily close to <math display="inline">1</math>. We say that "the limit of the sequence <math display="inline">n \times \sin\left(\tfrac1{n}\right)</math> equals <math display="inline">1</math>."

</div>

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math>\lim_{n \to \infty}a_n</math>). If such a limit exists and is finite, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus 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."

Pietro Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometriae speciosae elementa (1659). He used the term quasi-infinite for unbounded and quasi-null for vanishing.

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 <math display="inline">(x+o)^n</math>, which he then linearizes by taking the limit as <math display="inline">o</math> tends to <math display="inline">0</math>.

In the 18th century, mathematicians such as 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 study of hypergeometric 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 <math display="inline">\varepsilon</math> there exists an index <math display="inline">N</math> so that ...) was given by Bernard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and by Karl Weierstrass in the 1870s.

Real numbers

320px|thumb|The plot of a convergent sequence {a<sub>n</sub>} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases.

In the real numbers, a number <math>L</math> is the limit of the sequence <math>(x_n)</math>, if the numbers in the sequence become closer and closer to <math>L</math>, and not to any other number.

Examples

Examples of limit of a sequence in real numbers are the following:

If <math>x_n = c</math> for constant <math display="inline">c</math>, then <math>x_n \to c</math>.
If <math>x_n = \frac{1}{n}</math>, then <math>x_n \to 0</math>.

In other words, for every measure of closeness <math>\varepsilon</math>, the sequence's terms are eventually that close to the limit. The sequence <math>(x_n)</math> is said to converge to or tend to the limit <math>x</math>.

Symbolically, this is:

:<math>\forall \varepsilon > 0 \left(\exists N \in \N \left(\forall n \in \N \left(n \geq N \implies |x_n - x| < \varepsilon \right)\right)\right)</math>.

If a sequence <math>(x_n)</math> converges to some limit <math>x</math>, then it is convergent and <math>x</math> is the only limit; otherwise <math>(x_n)</math> is divergent. A sequence that has zero as its limit is sometimes called a null sequence.

Illustration

File:Folgenglieder im KOSY.svg|Example of a sequence which converges to the limit <math>a</math>|alt=Example of a sequence which converges to the limit <nowiki> </nowiki> a <nowiki> </nowiki> {\displaystyle a} .

File:Epsilonschlauch.svg|Regardless which <math>\varepsilon > 0</math> we have, there is an index <math>N_0</math>, so that the sequence lies afterwards completely in the epsilon tube <math>(a-\varepsilon,a+\varepsilon)</math>.

File:Epsilonschlauch klein.svg|There is also for a smaller <math>\varepsilon_1 > 0</math> an index <math>N_1</math>, so that the sequence is afterwards inside the epsilon tube <math>(a-\varepsilon_1,a+\varepsilon_1)</math>.

File:Epsilonschlauch2.svg|For each <math>\varepsilon > 0</math> there are only finitely many sequence members outside the epsilon tube.

</gallery>

Properties

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

When it exists, the limit of a sequence is unique.

This coincides with the definition given for metric spaces, if <math>(X, d)</math> is a metric space and <math>\tau</math> is the topology generated by <math>d</math>.

A limit of a sequence of points <math>\left(x_n\right)_{n \in \N}</math> in a topological space <math>T</math> is a special case of a limit of a function: the domain is <math>\N</math> in the space <math>\N \cup \lbrace + \infty \rbrace</math>, with the induced topology of the affinely extended real number system, the range is <math>T</math>, and the function argument <math>n</math> tends to <math>+\infty</math>, which in this space is a limit point of <math>\N</math>.

Properties

In a Hausdorff space, limits of sequences are unique whenever they exist. This need not be the case in non-Hausdorff spaces; in particular, if two points <math>x</math> and <math>y</math> are topologically indistinguishable, then any sequence that converges to <math>x</math> must converge to <math>y</math> and vice versa.

Hyperreal numbers

The definition of the limit using the hyperreal 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 <math>(x_n)</math> tends to L if for every infinite hypernatural <math display="inline">H</math>, the term <math>x_H</math> is infinitely close to <math display="inline">L</math> (i.e., the difference <math>x_H - L</math> is infinitesimal). Equivalently, L is the standard part of <math>x_H</math>:

:<math> L = {\rm st}(x_H)</math>.

Thus, the limit can be defined by the formula

:<math>\lim_{n \to \infty} x_n= {\rm st}(x_H)</math>.

where the limit exists if and only if the righthand side is independent of the choice of an infinite <math display="inline">H</math>.

Sequence of more than one index

Sometimes one may also consider a sequence with more than one index, for example, a double sequence <math>(x_{n, m})</math>. This sequence has a limit <math>L</math> if it becomes closer and closer to <math>L</math> when both n and m becomes very large.

Example

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

Definition

We call <math>x</math> the double limit of the sequence <math>(x_{n, m})</math>, written

:<math>x_{n, m} \to x</math>, or

:<math>\lim_{\begin{smallmatrix}

n \to \infty \\ m \to \infty

\end{smallmatrix x_{n, m} = x</math>,

if the following condition holds:

:For each real number <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that, for every pair of natural numbers <math>n, m \geq N</math>, we have <math>|x_{n, m} - x| < \varepsilon</math>.

In other words, for every measure of closeness <math>\varepsilon</math>, the sequence's terms are eventually that close to the limit. The sequence <math>(x_{n, m})</math> is said to converge to or tend to the limit <math>x</math>.

Symbolically, this is:

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

The double limit is different from taking limit in n first, and then in m. The latter is known as iterated 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

A sequence <math>(x_{n,m})</math> is said to tend to infinity, written

:<math>x_{n,m} \to \infty</math>, or

:<math>\lim_{\begin{smallmatrix}

n \to \infty \\ m \to \infty

\end{smallmatrixx_{n,m} = \infty</math>,

if the following holds:

:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every pair of natural numbers <math>n,m \geq N</math>, we have <math>x_{n,m} > K</math>; that is, the sequence terms are eventually larger than any fixed <math>K</math>.

Symbolically, this is:

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

Similarly, a sequence <math>(x_{n,m})</math> tends to minus infinity, written

:<math>x_{n,m} \to -\infty</math>, or

:<math>\lim_{\begin{smallmatrix}

n \to \infty \\ m \to \infty

\end{smallmatrixx_{n,m} = -\infty</math>,

if the following holds:

:For every real number <math>K</math>, there is a natural number <math>N</math> such that for every pair of natural numbers <math>n,m \geq N</math>, we have <math>x_{n,m} < K</math>; that is, the sequence terms are eventually smaller than any fixed <math>K</math>.

Symbolically, this is:

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

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 <math>x_{n,m}=(-1)^{n+m}</math> provides one such example.

Pointwise limits and uniform limits

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

:<math>x_{n, m} \to y_m\quad \text{pointwise}</math>, or

:<math>\lim_{n \to \infty} x_{n, m} = y_m\quad \text{pointwise}</math>,

which means:

:For each real number <math>\varepsilon > 0</math> and each fixed natural number <math>m</math>, there exists a natural number <math>N(\varepsilon, m) > 0</math> such that, for every natural number <math>n \geq N</math>, we have <math>|x_{n, m} - y_m| < \varepsilon</math>.

Symbolically, this is:

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

When such a limit exists, we say the sequence <math>(x_{n, m})</math> converges pointwise to <math>(y_m)</math>.

The second one is called uniform limit, denoted

:<math>x_{n, m} \to y_m \quad \text{uniformly}</math>,

:<math>\lim_{n \to \infty} x_{n, m} = y_m \quad \text{uniformly}</math>,

:<math>x_{n, m} \rightrightarrows y_m </math>, or

:<math>\underset{n\to\infty}{\mathrm{unif} \lim} \; x_{n, m} = y_m </math>,

which means:

:For each real number <math>\varepsilon > 0</math>, there exists a natural number <math>N(\varepsilon) > 0</math> such that, for every natural number <math>m</math> and for every natural number <math>n \geq N</math>, we have <math>|x_{n, m} - y_m| < \varepsilon</math>.