# Definition:Real Sequence

## Definition

A **real sequence** is a sequence (usually infinite) whose codomain is the set of real numbers $\R$.

### Notation

The notation for a **real sequence** is conventionally of the form $\sequence {a_n}$, where it is understood that:

- the domain of $\sequence {a_n}$ is the natural numbers: either $\set {0, 1, 2, 3, \ldots}$ or $\set {1, 2, 3, \ldots}$
- the range of $\sequence {a_n}$ is the set of real numbers $\R$.

However, some sources use the notation of mappings to explicitly interpret a **real sequence** as a real-valued function:

- $f: \N \to \R$

Hence the $n$th term can be seen denoted in two ways:

- $\sequence {a_n}$
- $\map f n$

## Examples

### Example: $\sequence {\paren {-1}^n}$

The first few terms of the real sequence:

- $S = \sequence {\paren {-1}^n}_{n \mathop \ge 1}$

are:

- $-1, +1, -1, +1, \dotsc$

This is an example of the real sequence:

- $S = \sequence {x^n}$

where $x = -1$.

$S$ is not monotone, either increasing or decreasing.

### Example: $\sequence {\dfrac {\paren {-1}^{n + 1} } n}$

The first few terms of the real sequence:

- $S = \sequence {\dfrac {\paren {-1}^{n + 1} } n}$

are:

- $1, -\dfrac 1 2, \dfrac 1 3, -\dfrac 1 4, \dotsc$

### Example: $\sequence {n^{-1} }$

The first few terms of the real sequence:

- $S = \sequence {n^{-1} }_{n \mathop \ge 1}$

are:

- $1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dotsc$

$S$ is strictly decreasing.

### Example: $\sequence 1$

The first few terms of the real sequence:

- $S = \sequence 1_{n \mathop \ge 1}$

are:

- $1, 1, 1, 1, \dotsc$

$S$ is both increasing and decreasing.

### Example: $\sequence {2^n}$

The first few terms of the real sequence:

- $S = \sequence {2^n}_{n \mathop \ge 1}$

are:

- $2, 4, 8, 16, \dotsc$

$S$ is strictly increasing.

### Example: $\sequence {x^n}$

The first few terms of the real sequence:

- $S = \sequence {x^n}$

are:

- $x, x^2, x^3, \ldots$

### Example: $\sequence {n^s}$

Let $s$ be a constant.

The first few terms of the real sequence:

- $S = \sequence {n^s}$

are:

- $1^s, 2^s, 3^s, \ldots$

When $s = 1$, $S$ is the sequence of natural numbers.

### Example: $\sequence {\dfrac 1 2 \paren {x_{n - 1} + \dfrac 2 {x_{n - 1} } } }_{n \mathop \ge 2}$

The first few terms of the real sequence:

- $S = \sequence {a_n}_{n \mathop \ge 1}$

defined as:

- $a_n = \begin {cases} 2 & : n = 1 \\ \dfrac 1 2 \paren {x_{n - 1} + \dfrac 2 {x_{n - 1} } } & : n > 1 \end {cases}$

are:

- $2, \dfrac 3 2, \dfrac {17} {12}, \dfrac {577} {408}, \dotsc$

## Also see

- Results about
**real sequences**can be found here.

## Sources

- 1958: J.A. Green:
*Sequences and Series*... (next): Chapter $1$: Sequences: $1$. Infinite Sequences - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences