In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$. For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$. (a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$. (b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$. (c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$. (d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that $$x _ { n } - \alpha = \mathrm { O } \left( e ^ { s \phi ^ { n } } \right)$$
In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$.\\
For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$.\\
(a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$.\\
(b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$.\\
(c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$.\\
(d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that
$$x _ { n } - \alpha = \mathrm { O } \left( e ^ { s \phi ^ { n } } \right)$$