# 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: Arbitrary Sequence $2$

$s_n = \begin {cases} 1 & : n = 1 \\ 0 & : n = 2 \\ \dfrac {s_{n - 2} + s_{n - 1} } 2 & : n > 2 \end {cases}$

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 $2$.