Monotone Real Sequence/Examples
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Examples of Monotone Real Sequence
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: $\sequence {x_n}: x_{n + 1} = \dfrac 2 {x_n + 1}$
Let $\sequence {x_n}$ denote the real sequence defined as:
- $x_n = \begin {cases} a : 0 < a < 1 & : n = 1 \\ \dfrac 2 {x_n + 1} & : n > 1 \end {cases}$
Then the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n + 1} }$ are both monotone:
- $\sequence {2 n}$ is strictly decreasing
- $\sequence {2 n + 1}$ is strictly increasing
Hence $\sequence {x_n} \to 1$ as $n \to \infty$.