In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of ${\displaystyle a_{n}}$, ${\displaystyle b_{n}}$ and ${\displaystyle c_{n}}$, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence ${\displaystyle F}$ is generally denoted as ${\displaystyle F_{n}}$.

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded.

Examples and notation

A sequence can be thought of as a list of elements with a particular order.^[1][2] Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and 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 numbers.

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

A tiling with squares whose sides are successive Fibonacci numbers in length.

The prime numbers are the natural numbers greater than 1 that have no divisors 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 whose elements are 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, ...).^[1]

Other examples of sequences include those made up of rational numbers, real numbers 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 of rational numbers (e.g. via its decimal expansion). As another example, π 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 π, 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.

Another example of sequences is a sequence of functions, where each member of the sequence is a function whose shape is determined by a natural number indexing that function.

The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences.^[3]

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 π. One such notation is to write down a general formula for computing the nth 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 ${\displaystyle (2n)_{n\in \mathbb {N} }}$. The sequence of squares could be written as ${\displaystyle (n^{2})_{n\in \mathbb {N} }}$. The variable n is called an 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 ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$, which denotes a sequence whose nth element is given by the variable ${\displaystyle a_{n}}$. For example:

${\displaystyle {\begin{aligned}a_{1}&=1{\text{st element of }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{nd element }}\\a_{3}&=3{\text{rd element }}\\&\;\;\vdots \\a_{n-1}&=(n-1){\text{th element}}\\a_{n}&=n{\text{th element}}\\a_{n+1}&=(n+1){\text{th element}}\\&\;\;\vdots \end{aligned}}}$

One can consider multiple sequences at the same time by using different variables; e.g. ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ could be a different sequence than ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$. One can even consider a sequence of sequences: ${\displaystyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }}$ denotes a sequence whose mth term is the sequence ${\displaystyle (a_{m,n})_{n\in \mathbb {N} }}$.

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 ${\displaystyle (k^{2})_{k=1}^{10}}$ denotes the ten-term sequence of squares ${\displaystyle (1,4,9,\ldots ,100)}$. The limits ${\displaystyle \infty }$ and ${\displaystyle -\infty }$ 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 ${\displaystyle (a_{n})_{n=1}^{\infty }}$ is the same as the sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$, and does not contain an additional term "at infinity". The sequence ${\displaystyle (a_{n})_{n=-\infty }^{\infty }}$ is a bi-infinite sequence, and can also be written as ${\displaystyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )}$.

In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ${\displaystyle (a_{k})}$ 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

${\displaystyle (a_{k})_{k=0}^{\infty }=(a_{0},a_{1},a_{2},\ldots ).}$

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 numbers could be denoted in any of the following ways.

• ${\displaystyle (1,9,25,\ldots )}$
• ${\displaystyle (a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}}$
• ${\displaystyle (a_{2k-1})_{k=1}^{\infty },\qquad a_{k}=k^{2}}$
• ${\displaystyle (a_{k})_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}$
• ${\displaystyle \left((2k-1)^{2}\right)_{k=1}^{\infty }}$

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 ${\displaystyle (a_{k})_{k=1}^{\infty }}$, but it is not the same as the sequence denoted by the expression.

Defining a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often defined using 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

${\displaystyle a_{n}=a_{n-1}+a_{n-2},}$

with initial terms ${\displaystyle a_{0}=0}$ and ${\displaystyle a_{1}=1}$. 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,^[4] defined by the recurrence relation

${\displaystyle {\begin{cases}a_{n}=a_{n-1}-n,\quad {\text{if the result is positive and not already in the previous terms,}}\\a_{n}=a_{n-1}+n,\quad {\text{otherwise}},\end{cases}}}$

with initial term ${\displaystyle a_{0}=0.}$

A linear recurrence with constant coefficients is a recurrence relation of the form

${\displaystyle a_{n}=c_{0}+c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}$

where ${\displaystyle c_{0},\dots ,c_{k}}$ are constants. There is a general method for expressing the general term ${\displaystyle a_{n}}$ of such a sequence as a function of n; see Linear recurrence. In the case of the Fibonacci sequence, one has ${\displaystyle c_{0}=0,c_{1}=c_{2}=1,}$ and the resulting function of n is given by Binet's formula.

A holonomic sequence is a sequence defined by a recurrence relation of the form

${\displaystyle a_{n}=c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}$

where $$Zdroj:https://en.wikipedia.org?pojem=Sequential
>Text je dostupný pod licencí Creative Commons Uveďte autora – Zachovejte licenci, případně za dalších podmínek. Podrobnosti naleznete na stránce Podmínky užití.