Definition:Sequence

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Informal Definition

A sequence is a set of objects which is listed in a specific order, one after another.

Thus one can identify the elements of a sequence as being the first, the second, the third, ... the $n$th, and so on.


Formal Definition

A sequence is a mapping whose domain is a subset of the set of natural numbers $\N$.


It can be seen that a sequence is an instance of a family of elements indexed by $\N$.


Notation

The notation for a sequence is as follows.

If $f: A \to S$ is a sequence, then a symbol, for example "$a$", is chosen to represent elements of this sequence.

Then for each $k \in A$, $f \left({k}\right)$ is denoted $a_k$, and $f$ itself is denoted $\left \langle {a_k} \right \rangle_{k \mathop \in A}$.

Other types of brackets may be encountered, for example:

$\left({a_k}\right)_{k \mathop \in A}$
$\left\{{a_k}\right\}_{k \mathop \in A}$

The latter is discouraged because of the implication that the order of the terms does not matter.


Any expression can be used to denote the domain of $f$ in place of $k \in A$.

For example:

$\left \langle {a_k} \right \rangle_{k \mathop \ge n}$
$\left \langle {a_k} \right \rangle_{p \mathop \le k \mathop \le q}$

The sequence itself may be defined by a simple formula, and so for example:

$\left \langle {k^3} \right \rangle_{2 \mathop \le k \mathop \le 6}$

is the same as:

$\left \langle {a_k} \right \rangle_{2 \mathop \le k \mathop \le 6}$ where $a_k = k^3$ for all $k \in \left\{{2, 3, \ldots, 6}\right\}$.


The set $A$ is usually taken to be the set of natural numbers $\N = \left\{{0,1, 2, 3, \ldots}\right\}$ or a subset.

In particular, for a finite sequence, $A$ is usually $\left\{{0, 1, 2, \ldots, n-1}\right\}$ or $\left\{{1, 2, 3, \ldots, n}\right\}$.

If this is the case, then it is usual to write $\left \langle {a_k} \right \rangle_{k \mathop \in A}$ as $\left \langle {a_k} \right \rangle$ or even as $\left \langle {a} \right \rangle$ if brevity and simplicity improve clarity.


A finite sequence of length $n$ can be denoted:

$\left({a_1, a_2, \ldots, a_n}\right)$

and by this notational convention the brackets are always round.


Terms

The elements of a sequence are known as its terms.


Let $\sequence {x_n}$ be a sequence.

Then the $k$th term of $\sequence {x_n}$ is the ordered pair $\tuple {k, x_k}$.


Finite Sequence

A finite sequence is a sequence whose domain is finite.


Length of Sequence

The length of a finite sequence is the number of terms it contains, or equivalently, the cardinality of its domain.


Sequence of $n$ Terms

A sequence of $n$ terms is a (finite) sequence whose length is $n$.


Empty Sequence

An empty sequence is a (finite) sequence containing no terms.

Thus an empty sequence is a mapping from $\varnothing$ to $S$, that is, the empty mapping.


Infinite Sequence

An infinite sequence is a sequence whose domain is infinite.


Range

Let $\sequence {x_n}_{n \mathop \in A}$ be a sequence.


The range of $\sequence {x_n}$ is the set:

$\set {x_n: n \mathop \in A}$


Codomain

The codomain of a sequence can be elements of a set of any objects.

If the codomain of a sequence $f$ is $S$, then the sequence is said to be a sequence of elements of $S$, or a sequence in $S$.


Sequence of Distinct Terms

A sequence of distinct terms of $S$ is an injection from a subset of $\N$ into $S$.

Thus a sequence $\sequence {a_k}_{k \mathop \in A}$ is a sequence of distinct terms if and only if:

$\forall j, k \in A: j \ne k \implies a_j \ne a_k$


Equality of Sequences

Let $f$ and $g$ be two sequences on the same set $A$:

$f = \left\langle{a_k}\right\rangle_{k \mathop \in A}$
$g = \left\langle{b_k}\right\rangle_{k \mathop \in B}$

Then $f = g$ if and only if:

$A = B$
$\forall i \in A: a_i = b_i$


Extension of Sequence

As a sequence is, by definition, also a mapping, the definition of an extension of a sequence is the same as that for an extension of a mapping:

Let:

$\left \langle {a_k} \right \rangle_{k \mathop \in A}$ be a sequence on $A$, where $A \subseteq \N$.
$\left \langle {b_k} \right \rangle_{k \mathop \in B}$ be a sequence on $B$, where $B \subseteq \N$.
$A \subseteq B$
$\forall k \in A: b_k = a_k$.

Then $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ extends or is an extension of $\left \langle {a_k} \right \rangle_{k \mathop \in A}$.


Negative Integers

A sequence on $\N$ can be extended to the negative integers.

Let $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ and $\left \langle {b_k} \right \rangle_{k \mathop \in \N}$ be sequences on $\N$.

Let $a_0 = b_0$.


Let $c_k$ be defined as:

$\forall k \in \Z: c_k = \begin{cases} a_k & : k \ge 0 \\ b_{-k} : & k \le 0 \end{cases}$


Then $\left \langle {c_k} \right \rangle_{k \mathop \in \Z}$ extends (or is an extension of) $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ to the negative integers.


Doubly Subscripted Sequence

A doubly subscripted sequence is a mapping whose domain is a subset of the cartesian product $\N \times \N$ of the set of natural numbers $\N$ with itself.


It can be seen that a doubly subscripted sequence is an instance of a family of elements indexed by $\N^2$.


A doubly subscripted sequence can be denoted $\left\langle{a_{m n} }\right\rangle_{m, \, n \mathop \ge 0}$


Also defined as

Some sources, generally expositions of set theory, define a sequence as a mapping whose domain is an ordinal.

In such cases, the natural numbers $\N$ are defined (usually) by the von Neumann construction, resulting in the fact that the two definitions are in complete agreement.

Note, however, that this definition of sequence extends to the transfinite ordinals.

Some sources define a sequence as a succession of numbers only, particularly those addressing analysis.


Also known as

Some sources refer to a sequence as a series.

This usage is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$, as that term is used to mean something different.


Also see

  • Results about sequences can be found here.


Sources