Definition:Sequence

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

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 $$S$$.

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 $$k \in A$$.

If the set $$A$$ is understood, 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.

Terms
The elements of a sequence are known as its terms.

Sequence of Distinct Terms
A sequence of distinct terms of $$S$$ is an injection from a subset of $$\mathbb{N}$$ 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 $$j \ne k$$.

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

Finite Sequence
A finite sequence is a sequence whose domain is finite.

Infinite Sequence
An infinite sequence is a sequence whose domain is infinite.

Comment
Notation varies. A common notation is $$\left({a_k}\right)_{k \in A}$$ for $$\left \langle {a_k} \right \rangle_{k \in A}$$.