11.1: Sequences

Definition: counting set

the set

$$\begin{equation*} \mathbb{N}_{<m} = \{n \in \mathbb{N} \vert n \lt m \} = \{ 0,\, 1,\, \ldots ,\, m-1\} \end{equation*}$$

The set $\mathbb{N}_{<m}$ has exactly $m$ elements in it. In Chapter 12 we will use these counting sets to, well, count the elements in other sets. For now, we will use them to index the objects in an ordered list.

Definition: Finite sequence(from a set $A$)

a function $\mathbb{N}_{<m} \to A$

Definition: infinite sequence (from a set $A$)

a function $\mathbb{N} \to A$

Definition: term in a sequence

one of the image elements of the function defining the sequence

Definition: $a_k$

the $k^{th}$ term in a sequence, so that if $f: \mathbb{N}_{<m} \rightarrow A$ or $f: \mathbb{N} \rightarrow A$ is a sequence then $a_k = f(k)$

Definition: $\{a_k\}$

the collection of all terms in a sequence

Definition: $\{a_k\}_0^m$

the collection of the terms in a sequence up to (and including) the $m^{th}$ term (if the sequence is finite, this could represent all terms in the sequence for the appropriate $m$ value)

Definition: $\{a_k\}_0^\infty$

the collection of all terms in a sequence, where we are explicit that it is an infinite sequence

Remark

• Of course, we do not restrict ourselves to the letter $a$ to represent the terms of a sequence. We might write $b_k\text{,}$ or $s_k\text{,}$ etc..
• While we use set-like notation $\{\}$ to represent the collection of all terms in a sequence, this collection is not a set, since order and repetition matter.

Example $\PageIndex{1}$: Sequence of squares.

The sequence $\{k^2\}$ has terms $0, \, 1, \, 4, \, 9, \, 16, \, 25, \, \ldots , \, k^2, \, \ldots \text{.}$

Example $\PageIndex{2}$: Sequence of definite integral values.

The sequence

$$\begin{equation*} \left\{ (-1)^k\int_1^{k+1} \dfrac{dx}{x} \right\} \end{equation*}$$
has terms $0, \, -\ln 2, \, \ln 3, \, -\ln 4, \, \ldots , \, (-1)^k\,\ln (k+1), \, \ldots \text{.}$