Definition:Sequence

From ProofWiki

[edit] 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.

[edit] Formal Definition

A sequence is a mapping whose domain is a subset of $\N$ .

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

[edit] Terms

The elements of a sequence are known as its terms.

[edit] Finite Sequence

A finite sequence is a sequence whose domain is finite.

[edit] Length of a Sequence

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

[edit] Sequence of n Terms

A sequence of $n$ terms is a sequence whose domain has $n$ elements.

Such a sequence is also known as an ordered n-tuple.

[edit] Null Sequence

A null sequence (or empty sequence) is one containing no terms.

Thus it is a mapping from $\varnothing$ to $S$ and therefore is null.

[edit] Infinite Sequence

An infinite sequence is a sequence whose domain is infinite.

[edit] 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 $S$ .

[edit] Rational Sequence

A rational sequence is a (usually) infinite sequence whose codomain is the set of rational numbers $\Q$ .

[edit] Real Sequence

A real sequence is a (usually) infinite sequence whose codomain is the set of real numbers $\R$ .

[edit] 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 $\left \langle {a_k} \right \rangle_{k \in A}$ .

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

The set $A$ is usually understood to be the set $\left\{{1, 2, 3, \ldots, n}\right\}$ .

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

[edit] Sequence of Distinct Terms

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

A sequence $\left \langle {a_k} \right \rangle_{k \in A}$ is a sequence of distinct terms iff $a_j \ne a_k$ for all $j, k \in A$ such that $j \ne k$ .

[edit] Equality of Sequences

Let $f$ and $g$ be two sequences:

$f = \left({x_1, x_2, \ldots, x_n}\right)$
$g = \left({y_1, y_2, \ldots, y_m}\right)$

Then $f = g$ iff:

$m = n$ ;
$\forall i: 1 \le i \ne n: x_i = y_i$

[edit] Notational Variants

Notation varies. Common variants for $\left \langle {a_k} \right \rangle$ are:

$\left({a_k}\right)$ ;
$\left\{{a_k}\right\}$ (this one is not recommended though, because of the implication that the order of the terms does not matter).