Real Sequence/Examples
Examples of Real Sequences
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:
- $2, \dfrac 3 2, \dfrac {17} {12}, \dfrac {577} {408}, \dotsc$
Example: $\sequence {\dfrac n {n + 1} }$
The real sequence $S$ whose first few terms are:
- $\dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dotsc$
can be defined by the formula:
- $S = \sequence {\dfrac n {n + 1} }_{n \mathop \ge 1}$
$S$ is strictly increasing.
Example: $\sequence {\paren {-1}^n 2^{-n} }$
The real sequence whose first few terms are:
- $1, -\dfrac 1 2, \dfrac 1 4, -\dfrac 1 8, \dotsc$
can be defined by the formula:
- $S = \sequence {\paren {-1}^n \dfrac 1 {2^n} }_{n \mathop \ge 0}$
Example: Arbitrary Sequence $1$
Consider the real sequence whose first few terms are:
- $\dfrac 1 2, 1, -\dfrac 1 2, -1, \dfrac 1 4, \dfrac 1 2, \dotsc$
has no obvious formula to define it.
Example: $n$
The real sequence whose first few terms are:
- $1, 2, 3, \dotsc$
can be defined by the formula:
- $S = \sequence n_{n \mathop \ge 1}$
Example: $\dfrac {1 + \paren {-1}^n} 2$
The real sequence whose first few terms are:
- $1, 0, 1, 0, \dotsc$
can be defined by the formula:
- $S = \sequence {\dfrac {1 + \paren {-1}^n} 2}_{n \mathop \ge 0}$
Example: $x^n$ for $0 < x < 1$
Real Sequence/Examples/x^n for 0 lt x lt 1
Example: Arbitrary Sequence $2$
The recurrence relation:
- $s_n = \begin {cases} 1 & : n = 1 \\
0 & : n = 2 \\ \dfrac {s_{n - 2} + s_{n - 1} } 2 & : n > 2 \end {cases}$
defines a real sequence whose first few terms are:
- $1, 0, \dfrac 1 2, \dfrac 1 4, \dfrac 3 8, \dfrac 5 {16}, \dotsc$
Example: $\sqrt {2 + \sqrt {x_n} }$
Let $\sequence {x_n}$ denote the real sequence defined as:
- $x_n = \begin {cases} \sqrt 2 : n = 1 \\ \sqrt {2 + \sqrt {x_{n - 1} } } & : n > 1 \end {cases}$
Then $\sequence {x_n}$ converges to a root of $x^4 - 4 x^2 - x + 4 = 0$ between $\sqrt 3$ and $2$.