Definition:Sequence

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[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 "nth", and so on.


[edit] Formal Definition

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


[edit] Range

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


If the range 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] 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 ak, 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] Terms

The elements of a sequence are known as its terms.


[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] 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.


[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] Rational Sequence

A rational sequence is a (usually) infinite sequence of rational numbers.


[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).
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