We consider the function $f$ defined on the interval $[0;+\infty[$ by $$f(x) = 5 - \frac{4}{x+2}$$ It will be admitted that $f$ is differentiable on the interval $[0;+\infty[$. The curve $\mathscr{C}$ representing $f$ and the line $\mathscr{D}$ with equation $y = x$ have been drawn in an orthonormal coordinate system in Appendix 1.

1. Prove that $f$ is increasing on the interval $[0;+\infty[$.
2. Solve the equation $f(x) = x$ on the interval $[0;+\infty[$. We denote the solution by $\alpha$. The exact value of $\alpha$ will be given, then an approximate value to $10^{-2}$ will be given.
3. We consider the sequence $(u_{n})$ defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = f(u_{n})$.
   On the figure in Appendix 1, using the curve $\mathscr{C}$ and the line $\mathscr{D}$, place the points $M_{0}$, $M_{1}$ and $M_{2}$ with zero ordinate and abscissae $u_{0}$, $u_{1}$ and $u_{2}$ respectively. What conjectures can be made about the direction of variation and the convergence of the sequence $(u_{n})$?
4. a. Prove, by induction, that for every natural integer $n$, $$0 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha$$ where $\alpha$ is the real number defined in question 2. b. Can we affirm that the sequence $(u_{n})$ is convergent? The answer will be justified.
5. For every natural integer $n$, we define the sequence $(S_{n})$ by $$S_{n} = \sum_{k=0}^{n} u_{k} = u_{0} + u_{1} + \cdots + u_{n}$$ a. Calculate $S_{0}$, $S_{1}$ and $S_{2}$. Give an approximate value of the results to $10^{-2}$ near. b. Complete the algorithm given in Appendix 2 so that it displays the sum $S_{n}$ for the value of the integer $n$ requested from the user. c. Show that the sequence $(S_{n})$ diverges to $+\infty$.

Consider the function $f$ defined on $]-1.5; +\infty[$ by $$f(x) = \ln(2x + 3) - 1$$ The purpose of this exercise is to study the convergence of the sequence $(u_{n})$ defined by: $$u_{0} = 0 \text{ and } u_{n+1} = f(u_{n}) \text{ for all natural integer } n.$$
Part A: Study of an auxiliary function Consider the function $g$ defined on $]-1.5; +\infty[$ by $g(x) = f(x) - x$.

1. Determine the limit of the function $g$ at $-1.5$.

We admit that the limit of the function $g$ at $+\infty$ is $-\infty$.

1. Study the variations of the function $g$ on $]-1.5; +\infty[$.
2. a. Prove that, in the interval $]-0.5; +\infty[$, the equation $g(x) = 0$ admits a unique solution $\alpha$. b. Determine an interval containing $\alpha$ with amplitude $10^{-2}$.

Part B: Study of the sequence $(u_{n})$ We admit that the function $f$ is strictly increasing on $]-1.5; +\infty[$.

1. Let $x$ be a real number. Show that if $x \in [-1; \alpha]$ then $f(x) \in [-1; \alpha]$.
2. a. Prove by induction that for all natural integer $n$: $$-1 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha.$$ b. Deduce that the sequence $(u_{n})$ converges.

9. Executing the flowchart below, if the input $\varepsilon$ is 0.01 , then the output value of $s$ equals [Figure]
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$

9. Executing the flowchart below, if the input $\varepsilon$ is 0.01 , then the output value of $S$ equals [Figure]
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$

We consider $I$ a real interval of strictly positive length, $f$ a function defined on $I$ with values in $I$ and $\left(u_{n}\right)_{n \in \mathbb{N}}$ a sequence defined by $u_{0} \in I$ and $\forall n \in \mathbb{N}, u_{n+1}=f\left(u_{n}\right)$. We assume that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ converges to an element $\ell$ of $I$ and that $f$ is differentiable at $\ell$.
a) Show that $f(\ell)=\ell$.
b) Show that if the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary then it belongs to $E^{c}$. Give its convergence rate as a function of $f^{\prime}(\ell)$.
c) Show that if $\left|f^{\prime}(\ell)\right|>1$, then $\left(u_{n}\right)_{n \in \mathbb{N}}$ is stationary.
d) Let $r$ be an integer greater than or equal to 2. We assume that the function $f$ is of class $\mathcal{C}^{r}$ on $I$ and that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary. Show that the convergence rate of $\left(u_{n}\right)_{n \in \mathbb{N}}$ is of order $r$ if and only if $\forall k \in\{1,2, \ldots, r-1\}, f^{(k)}(\ell)=0$.

If $\phi : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathcal { C } ^ { 1 }$ and satisfies $$\sup \left\{ \left| \phi ^ { \prime } ( x ) \right| ; x \in \mathbb { R } \right\} < 1$$ show that $\phi$ has at least one fixed point (one may study the sign of $x - \phi ( x )$ for $| x |$ sufficiently large). Show that this fixed point is unique.

By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \quad \left| \phi ^ { \prime } ( x ) \right| < 1 .$$

Let $\ell$ be a strictly positive integer. Let $F$ be a closed subset of $\mathbb { R } ^ { \ell }$ and let $\phi : F \rightarrow F$ be a map. We assume that there exists $k \in [ 0,1 [$ such that $$\forall x \in F , \quad \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$ (a) We choose a point $x _ { 0 } \in F$. Show that the formula $x _ { n + 1 } = \phi \left( x _ { n } \right)$ defines a sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ of elements of $F$, and that this sequence is convergent in $F$.
(b) Deduce that $\phi$ has a unique fixed point in $F$.
(c) This fixed point being denoted $x ^ { * }$, bound $\left\| x _ { n } - x ^ { * } \right\|$ as a function of $\left\| x _ { 0 } - x ^ { * } \right\|$.
(d) In what precedes, we assume that $$\phi = \underbrace { \theta \circ \cdots \circ \theta } _ { m \text { times } } ,$$ where $\theta : F \rightarrow F$ is a map and $m \geqslant 2$ is an integer. Show that $\theta$ has a fixed point, and a unique one, in $F$.

Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$. We suppose only $\tau > 0$. Show that for all $x_0 \in \mathbb{R}$, the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ converges to a minimizer of $f$.