Definition:Sequence/Notation

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 $k \in A$, $\map f k$ 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.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations