Convergent Real Sequence/Examples/x (n+1) = x n^2 + k
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Example of Convergent Real Sequence
Let $\sequence {x_n}$ be the real sequence defined as:
- $x_n = \begin {cases} h & : n = 1 \\ {x_{n - 1} }^2 + k & : n > 1 \end {cases}$
where:
- $0 < k < \dfrac 1 4$
- $a < h < b$, where $a$ and $b$ are the roots of the quadratic equation: $x^2 - x + k = 0$.
Then $\sequence {x_n}$ is convergent such that:
- $\ds \lim_{n \mathop \to \infty} x_n = a$
Proof
First some lemmata:
Lemma 1
- $\forall n \in \N_{>0}: a < x_n < b$
$\Box$
Lemma 2
Let $a$ and $b$ be the roots of the quadratic equation:
- $(1): \quad x^2 - x + k = 0$
Let:
- $0 < k < \dfrac 1 4$
Then $a$ and $b$ are both strictly positive real numbers.
$\Box$
From Lemma 1 and Lemma 2 we have that:
- $0 < a < x_n < b$
for all $n \in \N_{>0}$.
Then:
\(\ds x_{n + 1} - x_n\) | \(=\) | \(\ds {x_n}^2 - x_n + k\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds 0\) | Sign of Quadratic Function Between Roots | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_{n + 1}\) | \(<\) | \(\ds x_n\) |
Hence:
- $0 < a < x_{n + 1} < x_n < b$
Thus $\sequence {x_n}$ is decreasing and bounded below by $a$.
Hence by the Monotone Convergence Theorem (Real Analysis), $\sequence {x_n}$ converges to its infimum.
It remains to be shown that $\map \inf {x_n} = a$.
Suppose that:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
Thus:
\(\ds \lim_{n \mathop \to \infty} x_n\) | \(=\) | \(\ds l\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{n \mathop \to \infty} x_{n + 1}\) | \(=\) | \(\ds l\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds l^2 + k\) | \(=\) | \(\ds l\) | as $x_{n + 1} = {x_n}^2 + k$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds l^2 - l + k\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds l\) | \(=\) | \(\ds a\) | |||||||||||
\(\, \ds \text {or} \, \) | \(\ds l\) | \(=\) | \(\ds b\) |
But $b$ cannot be the infimum of $\sequence {x_n}$ because it is not a lower bound.
Hence:
- $\ds \lim_{n \mathop \to \infty} x_n = a$
and the result follows.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (2)$