Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n
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Example of Convergent Real Sequence
Let $h, k \in \R_{>0}$.
Let $\sequence {x_n}$ be the real sequence defined as:
- $x_n = \begin {cases} h & : n = 1 \\ \dfrac k {1 + x_{n - 1} } & : n > 1 \end {cases}$
Then $\sequence {x_n}$ is convergent to the positive root of the quadratic equation:
- $x^2 + x = k$
Proof
First some lemmata:
Lemma 1
- $\forall n \in \N_{>1}: k > x_n > 0$
$\Box$
Lemma 2
Consider the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$.
One of them is strictly increasing and the other is strictly decreasing.
$\Box$
From Lemma 2, We have that both $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$ is strictly monotone (one strictly increasing and the other strictly decreasing).
From Lemma 1, they are both bounded above by $k$ and bounded below by $0$.
Hence from the Monotone Convergence Theorem (Real Analysis), they both converge.
Let:
- $x_{2 n} \to l$ as $n \to \infty$
- $x_{2 n - 1} \to m$ as $n \to \infty$
Then:
\(\ds l\) | \(=\) | \(\ds \dfrac k {1 + m}\) | ||||||||||||
\(\ds m\) | \(=\) | \(\ds \dfrac k {1 + l}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds l + l m\) | \(=\) | \(\ds k\) | |||||||||||
\(\ds m + l m\) | \(=\) | \(\ds k\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds l\) | \(=\) | \(\ds m\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds l^2 + l\) | \(=\) | \(\ds k\) |
Hence the result.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$