# Real Sequence/Examples

## Contents

- 1 Examples of Real Sequences
- 1.1 Example: $\sequence {\paren {-1}^n}$
- 1.2 Example: $\sequence {n^{-1} }$
- 1.3 Example: $\sequence 1$
- 1.4 Example: $\sequence {2^n}$
- 1.5 Example: $\sequence {\dfrac 1 2 \paren {x_{n - 1} + \dfrac 2 {x_{n - 1} } } }_{n \mathop \ge 2}$
- 1.6 Example: $\sequence {\dfrac n {n + 1} }$
- 1.7 Example: $\sequence {\paren {-1}^n 2^{-n} }$
- 1.8 Example: Arbitrary Sequence $1$
- 1.9 Example: $n$
- 1.10 Example: $\dfrac {1 + \paren {-1}^n} 2$
- 1.11 Example: Arbitrary Sequence $2$

## 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:

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

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

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

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 whose first few terms are:

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

can be defined by the formula:

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

### 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 $2$

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:

- $1, 0, \dfrac 1 2, \dfrac 1 4, \dfrac 3 8, \dfrac 5 {16}, \dotsc$