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