Definition:Sequence/Notation

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Definition

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.


Sources