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:

$\displaystyle \lim_{n \mathop \to \infty} x_n = a$


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:

\(\displaystyle x_{n + 1} - x_n\) \(=\) \(\displaystyle {x_n}^2 - x_n + k\)
\(\displaystyle \) \(<\) \(\displaystyle 0\) Sign of Quadratic Function Between Roots
\(\displaystyle \leadsto \ \ \) \(\displaystyle x_{n + 1}\) \(<\) \(\displaystyle x_n\)

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:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Thus:

\(\displaystyle \lim_{n \mathop \to \infty} x_n\) \(=\) \(\displaystyle l\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \lim_{n \mathop \to \infty} x_{n + 1}\) \(=\) \(\displaystyle l\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle l^2 + k\) \(=\) \(\displaystyle l\) as $x_{n + 1} = {x_n}^2 + k$
\(\displaystyle \leadsto \ \ \) \(\displaystyle l^2 - l + k\) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle l\) \(=\) \(\displaystyle a\)
\(\, \displaystyle \text {or} \, \) \(\displaystyle l\) \(=\) \(\displaystyle b\)


But $b$ cannot be the infimum of $\sequence {x_n}$ because it is not a lower bound.

Hence:

$\displaystyle \lim_{n \mathop \to \infty} x_n = a$

and the result follows.

$\blacksquare$


Sources