Editing Sequence (section)

==Examples and notation==
A sequence can be thought of as a list of elements with a particular order.<ref name=":0">{{Cite web|title=Sequences|url=https://www.mathsisfun.com/algebra/sequences-series.html|access-date=2020-08-17|website=www.mathsisfun.com|archive-date=2020-08-12|archive-url=https://web.archive.org/web/20200812220432/https://mathsisfun.com/algebra/sequences-series.html|url-status=live}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Sequence|url=https://mathworld.wolfram.com/Sequence.html|access-date=2020-08-17|website=mathworld.wolfram.com|language=en|archive-date=2020-07-25|archive-url=https://web.archive.org/web/20200725104417/https://mathworld.wolfram.com/Sequence.html|url-status=live}}</ref> Sequences are useful in a number of mathematical disciplines for studying [[Function (mathematics)|functions]], [[Space (mathematics)|spaces]], and other mathematical structures using the [[#Limits and convergence|convergence]] properties of sequences. In particular, sequences are the basis for [[series (mathematics)|series]], which are important in [[differential equations]] and [[analysis (mathematics)|analysis]]. Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of [[prime number]]s.

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with [[ellipsis]] leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting a sequence are discussed after the examples.

===Examples===
[[File:Fibonacci blocks.svg|thumb|A [[Tessellation|tiling]] with squares whose sides are successive Fibonacci numbers in length.]]

The [[prime number]]s are the [[natural numbers]] greater than 1 that have no [[divisor]]s but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in [[mathematics]], particularly in [[number theory]] where many results related to them exist.

The [[Fibonacci numbers]] comprise the integer sequence in which each element is the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).<ref name=":0" />

Other examples of sequences include those made up of [[rational numbers]], [[real number]]s and [[complex numbers]]. The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as the [[limit of a sequence|limit]] of a sequence of rational numbers (e.g. via its [[decimal expansion]], also see ''[[completeness of the real numbers]]''). As another example, [[pi|{{pi}}]] is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of {{pi}}, that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection.

Other examples are sequences of [[function (mathematics)|function]]s, whose elements are functions instead of numbers.

The [[On-Line Encyclopedia of Integer Sequences]] comprises a large list of examples of integer sequences.<ref>[https://oeis.org/wiki/Index_to_OEIS Index to OEIS] {{Webarchive|url=https://web.archive.org/web/20221018125328/https://oeis.org/wiki/Index_to_OEIS |date=2022-10-18 }}, On-Line Encyclopedia of Integer Sequences, 2020-12-03</ref>

===Indexing===
Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of [[pi|{{pi}}]]. One such notation is to write down a general formula for computing the ''n''th term as a function of ''n'', enclose it in parentheses, and include a subscript indicating the set of values that ''n'' can take. For example, in this notation the sequence of even numbers could be written as <math display=inline>(2n)_{n\in\mathbb N}</math>. The sequence of squares could be written as <math display=inline>(n^2)_{n\in\mathbb N}</math>. The variable ''n'' is called an [[Indexed family|index]], and the set of values that it can take is called the [[index set]].

It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like <math display=inline>(a_n)_{n\in\mathbb N}</math>, which denotes a sequence whose ''n''th element is given by the variable <math>a_n</math>. For example:
:<math>\begin{align}
a_1 &= 1\text{st element of }(a_n)_{n\in\mathbb N} \\
a_2 &= 2\text{nd element } \\
a_3 &= 3\text{rd element } \\
&\;\;\vdots \\
a_{n-1} &= (n-1)\text{th element} \\
a_n &= n\text{th element} \\
a_{n+1} &= (n+1)\text{th element} \\
&\;\; \vdots
\end{align}</math>
One can consider multiple sequences at the same time by using different variables; e.g. <math display=inline>(b_n)_{n\in\mathbb N}</math> could be a different sequence than <math display=inline>(a_n)_{n\in\mathbb N}</math>. One can even consider a sequence of sequences: <math display=inline>((a_{m, n})_{n\in\mathbb N})_{m\in\mathbb N}</math> denotes a sequence whose ''m''th term is the sequence <math display=inline>(a_{m, n})_{n\in\mathbb N}</math>.

An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation <math display=inline>(k^2){\vphantom)}_{k = 1}^{10}</math> denotes the ten-term sequence of squares <math>(1, 4, 9, \ldots, 100)</math>. The limits <math>\infty</math> and <math>-\infty</math> are allowed, but they do not represent valid values for the index, only the [[supremum]] or [[infimum]] of such values, respectively. For example, the sequence <math display=inline>{(a_n)}_{n = 1}^\infty</math> is the same as the sequence <math display=inline>(a_n)_{n\in\mathbb N}</math>, and does not contain an additional term "at infinity". The sequence <math display=inline>{(a_n)}_{n = -\infty}^\infty</math> is a '''bi-infinite sequence''', and can also be written as <math display=inline>(\ldots, a_{-1}, a_0, a_1, a_2, \ldots)</math>.

In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes <math display=inline>(a_k)</math> for an arbitrary sequence. Often, the index ''k'' is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in
:<math>{(a_k)}_{k=0}^\infty = ( a_0, a_1, a_2, \ldots ).</math>
In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of [[odd number]]s could be denoted in any of the following ways.

* <math>(1, 9, 25, \ldots)</math>
* <math>(a_1, a_3, a_5, \ldots), \qquad a_k = k^2</math>
* <math>{(a_{2k-1})}_{k=1}^\infty, \qquad a_k = k^2</math>
* <math>{(a_{k})}_{k=1}^\infty, \qquad a_k = (2k-1)^2</math>
* <math>\bigl((2k-1)^2\bigr)_{k=1}^\infty</math>

Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the [[natural numbers]]. In the second and third bullets, there is a well-defined sequence <math display=inline>{(a_{k})}_{k=1}^\infty</math>, but it is not the same as the sequence denoted by the expression.

===Defining a sequence by recursion===
{{main|Recurrence relation}}
Sequences whose elements are related to the previous elements in a straightforward way are often defined using [[Recursive definition|recursion]]. This is in contrast to the definition of sequences of elements as functions of their positions.

To define a sequence by recursion, one needs a rule, called ''recurrence relation'' to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation.

The [[Fibonacci sequence]] is a simple classical example, defined by the recurrence relation
:<math>a_n = a_{n-1} + a_{n-2},</math>
with initial terms <math>a_0 = 0</math> and <math>a_1 = 1</math>. From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34.

A complicated example of a sequence defined by a recurrence relation is [[Recamán's sequence]],<ref>{{cite OEIS|1=A005132|2=Recamán's sequence|access-date=26 January 2018}}</ref> defined by the recurrence relation
:<math>\begin{cases}a_n = a_{n-1} - n,\quad \text{if the result is positive and not already in the previous terms,}\\a_n = a_{n-1} + n, \quad\text{otherwise},
\end{cases}</math>
with initial term <math>a_0 = 0.</math>

A ''linear recurrence with constant coefficients'' is a recurrence relation of the form
:<math>a_n=c_0 +c_1a_{n-1}+\dots+c_k a_{n-k},</math>
where <math>c_0,\dots, c_k</math> are [[constant (mathematics)|constants]]. There is a general method for expressing the general term <math>a_n</math> of such a sequence as a function of {{mvar|n}}; see [[Linear recurrence]]. In the case of the Fibonacci sequence, one has <math>c_0=0, c_1=c_2=1,</math> and the resulting function of {{mvar|n}} is given by [[Binet's formula]].

A [[holonomic sequence]] is a sequence defined by a recurrence relation of the form 
:<math>a_n=c_1a_{n-1}+\dots+c_k a_{n-k},</math>
where <math>c_1,\dots, c_k</math> are [[polynomial]]s in {{mvar|n}}. For most holonomic sequences, there is no explicit formula for expressing <math>a_n</math> as a function of {{mvar|n}}. Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many [[special functions]] have a [[Taylor series]] whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions.

Not all sequences can be specified by a recurrence relation. An example is the sequence of [[prime number]]s in their natural order (2, 3, 5, 7, 11, 13, 17, ...).