Editing Sequence (section)

==Limits and convergence==
{{Main|Limit of a sequence}}

[[File:Converging Sequence example.svg|320px|thumb|The plot of a convergent sequence (''a<sub>n</sub>'') is shown in blue. From the graph we can see that the sequence is converging to the limit zero as ''n'' increases.]]

An important property of a sequence is ''convergence''. If a sequence converges, it converges to a particular value known as the ''limit''. If a sequence converges to some limit, then it is '''convergent'''. A sequence that does not converge is '''divergent'''.

Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value <math>L</math> (called the limit of the sequence), and they become and remain ''arbitrarily'' close to <math>L</math>, meaning that given a real number <math>d</math> greater than zero, all but a finite number of the elements of the sequence have a distance from <math>L</math> less than <math>d</math>.

For example, the sequence <math display="inline">a_n = \frac{n+1}{2n^2}</math> shown to the right converges to the value 0. On the other hand, the sequences <math display="inline">b_n = n^3</math> (which begins 1, 8, 27, ...) and <math>c_n = (-1)^n</math> (which begins −1, 1, −1, 1, ...) are both divergent.

If a sequence converges, then the value it converges to is unique. This value is called the '''limit''' of the sequence. The limit of a convergent sequence <math>(a_n)</math> is normally denoted <math display="inline">\lim_{n\to\infty}a_n</math>. If <math>(a_n)</math> is a divergent sequence, then the expression <math display="inline">\lim_{n\to\infty}a_n</math> is meaningless.

=== Formal definition of convergence===

A sequence of real numbers <math>(a_n)</math> '''converges to''' a real number <math>L</math> if, for all <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that for all <math>n \geq N</math> we have<ref name="Gaughan">{{cite book|title=Introduction to Analysis |last=Gaughan |first=Edward |publisher=AMS (2009)|isbn=978-0-8218-4787-9|chapter=1.1 Sequences and Convergence|year=2009 }}</ref> 
:<math>|a_n - L| < \varepsilon.</math>

If <math>(a_n)</math> is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that <math>|\cdot|</math> denotes the complex modulus, i.e. <math>|z| = \sqrt{z^*z}</math>. If <math>(a_n)</math> is a sequence of points in a [[metric space]], then the formula can be used to define convergence, if the expression <math>|a_n-L|</math> is replaced by the expression <math>\operatorname{dist}(a_n, L)</math>, which denotes the [[Metric (mathematics)|distance]] between <math>a_n</math> and <math>L</math>.

===Applications and important results===
If <math>(a_n)</math> and <math>(b_n)</math> are convergent sequences, then the following limits exist, and can be computed as follows:<ref name="Gaughan" /><ref name="Dawkins">{{cite web |url=http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx |title=Series and Sequences |last1=Dawikins |first1=Paul |work=Paul's Online Math Notes/Calc II (notes) |access-date=18 December 2012 |archive-date=30 November 2012 |archive-url=https://web.archive.org/web/20121130095834/http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx |url-status=live }}</ref>

* <math>\lim_{n\to\infty} (a_n \pm b_n) = \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math>
* <math>\lim_{n\to\infty} c a_n = c \lim_{n\to\infty} a_n</math> for all real numbers <math>c</math>
* <math>\lim_{n\to\infty} (a_n b_n) = \bigl( \lim_{n\to\infty} a_n \bigr) \bigl( \lim_{n\to\infty} b_n \bigr)</math>
* <math>\lim_{n\to\infty} \frac{a_n} {b_n} = \bigl( \lim \limits_{n\to\infty} a_n \bigr) \big/ \bigl( \lim \limits_{n\to\infty} b_n \bigr)</math>, provided that <math>\lim_{n\to\infty} b_n \ne 0</math>
* <math>\lim_{n\to\infty} a_n^p = \bigl( \lim_{n\to\infty} a_n \bigr)^p</math> for all <math>p > 0</math> and <math>a_n > 0</math>

Moreover:
* If <math>a_n \leq b_n</math> for all <math>n</math> greater than some <math>N</math>, then <math>\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n </math>.{{efn|If the inequalities are replaced by strict inequalities then this is false: There are sequences such that <math>a_n < b_n</math> for all <math>n</math>, but <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n </math>.}}
* ([[Squeeze Theorem]])<br>If <math>(c_n)</math> is a sequence such that <math>a_n \leq c_n \leq b_n</math> for all <math>n > N</math> {{nowrap|and <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L</math>,}}<br>then <math>(c_n)</math> is convergent, and <math>\lim_{n\to\infty} c_n = L</math>.
* If a sequence is [[#Bounded|bounded]] and [[#Increasing and decreasing|monotonic]] then it is convergent.
* A sequence is convergent if and only if all of its subsequences are convergent.

===Cauchy sequences===
{{Main|Cauchy sequence}}

[[File:Cauchy sequence illustration.svg|350px|thumb| The plot of a Cauchy sequence (''X<sub>n</sub>''), shown in blue, as ''X<sub>n</sub>'' versus ''n''. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as ''n'' increases. In the [[real number]]s every Cauchy sequence converges to some limit.]]

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in [[metric spaces]], and, in particular, in [[real analysis]]. One particularly important result in real analysis is ''Cauchy characterization of convergence for sequences'':
:A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.
In contrast, there are Cauchy sequences of [[rational numbers]] that are not convergent in the rationals, e.g. the sequence defined by <math>x_1 = 1</math> and <math>x_{n+1} = \tfrac12\bigl(x_n + \tfrac{2}{x_n}\bigr)</math> is Cauchy, but has no rational limit (cf. {{slink|Cauchy sequence#Non-example: rational numbers}}). More generally, any sequence of rational numbers that converges to an [[irrational number]] is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers.

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called [[complete metric space]]s and are particularly nice for analysis.

=== Infinite limits ===

In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If <math>a_n</math> becomes arbitrarily large as <math>n \to \infty</math>, we write
:<math>\lim_{n\to\infty}a_n = \infty.</math>
In this case we say that the sequence '''diverges''', or that it '''converges to infinity'''. An example of such a sequence is {{nowrap|1=''a''<sub>''n''</sub> = ''n''}}.

If <math>a_n</math> becomes arbitrarily negative (i.e. negative and large in magnitude) as <math>n \to \infty</math>, we write
:<math>\lim_{n\to\infty}a_n = -\infty</math>
and say that the sequence '''diverges''' or '''converges to negative infinity'''.