Exercise 4 (5 points) -- Candidates who have NOT chosen the specialization option
Part A
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 3u_{n}}{3 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.

1. Prove by induction that, for all natural integers $n$, we have: $u_{n} > 1$.
2. a. Establish that, for all natural integers $n$, we have: $u_{n+1} - u_{n} = \frac{(1 - u_{n})(1 + u_{n})}{3 + u_{n}}$. b. Determine the direction of variation of the sequence $(u_{n})$. Deduce that the sequence $(u_{n})$ converges.

Part B
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 0.5u_{n}}{0.5 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.

1. Consider the following algorithm:
   

   Input	Let $n$ be a non-zero natural integer
   Initialization	Assign to $u$ the value 2
   Processing	FOR $i$ going from 1 to $n$
   and	Assign to $u$ the value $\frac{1 + 0.5u}{0.5 + u}$
   output	Display $u$
   	END FOR

   
   Reproduce and complete the following table by running this algorithm for $n = 3$. The values of $u$ should be rounded to the nearest thousandth.
   

   $i$	1	2	3
   $u$

   
   
2. For $n = 12$, the previous table was extended and we obtained:
   

   $i$	4	5	6	7	8	9	10	11	12
   $u$	1.0083	0.9973	1.0009	0.9997	1.0001	0.99997	1.00001	0.999996	1.000001

   
   Conjecture the behavior of the sequence $(u_{n})$ at infinity.
3. Consider the sequence $(v_{n})$ defined, for all natural integers $n$, by: $v_{n} = \frac{u_{n} - 1}{u_{n} + 1}$. a. Prove that the sequence $(v_{n})$ is geometric with common ratio $-\frac{1}{3}$. b. Calculate $v_{0}$ then write $v_{n}$ as a function of $n$.
4. a. Show that, for all natural integers $n$, we have: $v_{n} \neq 1$. b. Show that, for all natural integers $n$, we have: $u_{n} = \frac{1 + v_{n}}{1 - v_{n}}$. c. Determine the limit of the sequence $(u_{n})$.

Let $a$ be a fixed non-zero real number. The purpose of this exercise is to study the sequence $(u_n)$ defined by: $$u_0 = a \quad \text{and, for all } n \text{ in } \mathbb{N}, \quad u_{n+1} = \mathrm{e}^{2u_n} - \mathrm{e}^{u_n}.$$ Note that this equality can also be written: $u_{n+1} = e^{u_n}(\mathrm{e}^{u_n} - 1)$.

1. Let $g$ be the function defined for all real $x$ by $$g(x) = \mathrm{e}^{2x} - \mathrm{e}^x - x.$$ a) Calculate $g'(x)$ and prove that, for all real $x$: $g'(x) = (\mathrm{e}^x - 1)(2\mathrm{e}^x + 1)$. b) Determine the variations of the function $g$ and give the value of its minimum. c) By noting that $u_{n+1} - u_n = g(u_n)$, study the direction of variation of the sequence $(u_n)$.
   
2. In this question, we assume that $a \leqslant 0$. a) Prove by induction that, for all natural integer $n$, $u_n \leqslant 0$. b) Deduce from the previous questions that the sequence $(u_n)$ is convergent. c) In the case where $a$ equals 0, give the limit of the sequence $(u_n)$.
   
3. In this question, we assume that $a > 0$.
   Since the sequence $(u_n)$ is increasing, question 1 allows us to assert that, for all natural integer $n$, $u_n \geqslant a$. a) Prove that, for all natural integer $n$, we have: $u_{n+1} - u_n \geqslant g(a)$. b) Prove by induction that, for all natural integer $n$, we have: $$u_n \geqslant a + n \times g(a).$$ c) Determine the limit of the sequence $(u_n)$.
   
4. In this question, we take $a = 0.02$.
   The following algorithm is intended to determine the smallest integer $n$ such that $u_n > M$, where $M$ denotes a positive real number. This algorithm is incomplete.
   

   Variables	$n$ is an integer, $u$ and $M$ are two real numbers
   Initialization	\begin{tabular}{l} $u$ takes the value 0.02
   $n$ takes the value 0
   Enter the value of $M$

   \hline Processing & While $\cdots$ & $\ldots$ & $\ldots$ End while & \end{tabular}
   a) On your paper, rewrite the ``Processing'' part by completing it. b) Using a calculator, determine the value that this algorithm will display if $M = 60$.

The purpose of Part A is to study the behavior of the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 0.3$ and by the recurrence relation, for all natural integer $n$ :
$$u _ { n + 1 } = 2 u _ { n } \left( 1 - u _ { n } \right)$$
This recurrence relation is written $u _ { n + 1 } = f \left( u _ { n } \right)$, where $f$ is the function defined on $\mathbb { R }$ by :
$$f ( x ) = 2 x ( 1 - x )$$

1. Prove that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$.
2. We admit that for all natural integer $n , 0 \leqslant u _ { n } \leqslant \frac { 1 } { 2 }$. Calculate $u _ { 1 }$ then perform a proof by induction to demonstrate that for all natural integer $n , u _ { n } \leqslant u _ { n + 1 }$.
3. Deduce that the sequence $( u _ { n } )$ is convergent.
4. Justify that the limit of the sequence $( u _ { n } )$ is equal to $\frac { 1 } { 2 }$.

We consider a function $f$ of class $\mathcal{C}^2$ on $[0,1]$ taking values in $[0,1]$ such that $f'$ and $f''$ take non-negative values. We assume $f(1)=1$, $f'(0)<1$ and $f''(1)>0$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
Show that the sequence $(u_n)_{n\in\mathbb{N}}$ is increasing, then that it is convergent. We denote its limit by $l$.

Let $\left(u_{n}\right)_{n \geqslant 0}$ be a real sequence such that: $\forall (m,n) \in \mathbb{N}^{2}, \quad u_{m+n} \geqslant u_{m} + u_{n}$. We suppose that the set $\left\{\frac{u_{n}}{n}, n \in \mathbb{N}^{*}\right\}$ is bounded above and we denote by $s$ its supremum.
Let $m$, $q$ and $r$ be elements of $\mathbb{N}$. We set $n = mq + r$. Compare the two real numbers $u_{n}$ and $qu_{m} + u_{r}$ and show that $u_{n} - ns \geqslant q\left(u_{m} - ms\right) + u_{r} - rs$.

Let $f$ be a function with real values, defined and continuous on $\mathbb{R}^{+}$, and admitting a finite limit at $+\infty$.
a) Show that $f$ is bounded on $\mathbb{R}^{+}$.
b) Deduce that the function $g$ defined on $\mathbb{R}^{+}$ by $\forall t \in \mathbb{R}^{+}, g(t)=t e^{\gamma t}$ where $\gamma$ is a strictly negative real, is bounded on $\mathbb{R}^{+}$.

Verify that a periodic sequence is bounded.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $| a | > 2$, the zero sequence is the only periodic solution of (II.1).

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = - 2$ then, (II.1) admits infinitely many constant solutions and infinitely many unbounded solutions.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = + 2$ then, (II.1) admits infinitely many 2-periodic solutions and infinitely many unbounded solutions.

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 in this question that $0 < x_0 < 1/\tau$. a) Justify that the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ is decreasing, with strictly positive values, and satisfies $x_{n+1} = x_n(1 - \tau x_n)$ for all $n \in \mathbb{N}$. b) Justify that $x_n \rightarrow 0$ when $n \rightarrow \infty$. c) Show that $1/x_{n+1} = 1/x_n + \tau/(1 - \tau x_n)$ for all $n \in \mathbb{N}$. Deduce that $x_n \leq x_0/(1 + n\tau x_0)$.

If $a _ { \alpha }$ is the greatest term in the sequence $a _ { n } = \frac { n ^ { 3 } } { n ^ { 4 } + 147 } , n = 1 , 2 , 3 \ldots$, then $\alpha$ is equal to $\_\_\_\_$