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 $\left \langle{x_n}\right \rangle$ be a sequence.

Then the $k$th term of $\left \langle{x_n}\right \rangle$ is the ordered pair $\left({k, x_k}\right)$.


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 $\left \langle{x_n}\right \rangle_{n \mathop \in A}$ be a sequence.


The range of $\left \langle{x_n}\right \rangle$ is the set:

$\left\{{x_n: n \in A}\right\}$


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 $\left \langle {a_k} \right \rangle_{k \in A}$ is a sequence of distinct terms iff:

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


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 as the finite ordinals, meaning the two definitions are in complete agreement.

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


Also see

  • Results about sequences can be found here.


Sources