Definition:Sequence
Definition
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 $k \in A$, $\map f k$ is denoted $a_k$, and $f$ itself is denoted $\sequence {a_k}_{k \mathop \in A}$.
Other types of brackets may be encountered, for example:
- $\tuple {a_k}_{k \mathop \in A}$
- $\set {a_k}_{k \mathop \in A}$
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:
- $\sequence {a_k}_{k \mathop \ge n}$
- $\sequence {a_k}_{p \mathop \le k \mathop \le q}$
The sequence itself may be defined by a simple formula, and so for example:
- $\sequence {k^3}_{2 \mathop \le k \mathop \le 6}$
is the same as:
- $\sequence {a_k}_{2 \mathop \le k \mathop \le 6}$ where $a_k = k^3$ for all $k \in \set {2, 3, \ldots, 6}$.
The set $A$ is usually taken to be the set of natural numbers $\N = \set {0,1, 2, 3, \ldots}$ or a subset.
In particular, for a finite sequence, $A$ is usually $\set {0, 1, 2, \ldots, n - 1}$ or $\set {1, 2, 3, \ldots, n}$.
If this is the case, then it is usual to write $\sequence {a_k}_{k \mathop \in A}$ as $\sequence {a_k}$ or even as $\sequence a$ if brevity and simplicity improve clarity.
A finite sequence of length $n$ can be denoted:
- $\tuple {a_1, a_2, \ldots, a_n}$
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 with a finite number of terms.
Empty Sequence
An empty sequence is a (finite) sequence containing no terms.
Thus an empty sequence is a mapping from $\O$ to $S$, that is, the empty mapping.
Infinite Sequence
An infinite sequence is a sequence whose domain is infinite.
That is, an infinite sequence is a sequence that has infinitely many terms.
Hence for an infinite sequence $\sequence {s_n}_{n \mathop \in \N}$ whose range is $S$, $\sequence {s_n}_{n \mathop \in \N}$ is an element of the set of mappings $S^{\N}$ from $\N$ to $S$.
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 $S$.
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:
- $A = B$
- $\forall i \in A: a_i = b_i$
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 $\sequence {a_{m n} }_{m, \, n \mathop \ge 0}$
Examples
Square Numbers
The sequence of square numbers, for $n \in \Z_{\ge 0}$, begins:
- $0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, \ldots$
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
- Definition:Integer Sequence
- Definition:Rational Sequence
- Definition:Real Sequence
- Definition:Complex Sequence
- Definition:Ordinal Sequence, where the domain is an ordinal, not necessarily $\N$ or a subset thereof
- Definition:Arithmetic Function, which can be considered as an example of an infinite sequence
- Results about sequences can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$: Example $13$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Example $6.1$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): sequence
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Sequences
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sequence