We denote by $\mathbb{C}$ the set of complex numbers. In the complex plane equipped with an orthonormal coordinate system $(O; \vec{u}, \vec{v})$ we have placed a point $M$ with affixe $z$ belonging to $\mathbb{C}$, then the point $R$ intersection of the circle with center $O$ passing through $M$ and the half-axis $[O; \vec{u})$.
Part A

1. Express the affixe of point $R$ as a function of $z$.
2. Let the point $M'$ with affixe $z'$ defined by $$z' = \frac{1}{2}\left(\frac{z + |z|}{2}\right).$$ Reproduce the figure on the answer sheet and construct the point $M'$.

Part B
We define the sequence of complex numbers $(z_n)$ by a first term $z_0$ belonging to $\mathbb{C}$ and, for every natural integer $n$, by the recurrence relation: $$z_{n+1} = \frac{z_n + |z_n|}{4}$$ The purpose of this part is to study whether the behavior at infinity of the sequence $(|z_n|)$ depends on the choice of $z_0$.

1. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a negative real number?
2. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a positive real number?
3. We now assume that $z_0$ is not a real number. a. What conjecture can we make about the behavior at infinity of the sequence $(|z_n|)$? b. Prove this conjecture, then conclude.

Let $(u_{n})$ be the sequence defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = u_{n} - \ln\left(u_{n}^{2} + 1\right)$.

1. Show by induction that, for every natural integer $n$, $u_{n}$ belongs to $[0;1]$.
2. Study the variations of the sequence $(u_{n})$.
3. Show that the sequence $(u_{n})$ is convergent.
4. We denote by $\ell$ its limit, and we admit that $\ell$ satisfies the equality $f(\ell) = \ell$. Deduce the value of $\ell$.

For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$'', with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$ for all $n \geqslant 1$. a. Prove by induction that, for all integer $n \geqslant 1$: $p_n > 0{,}8$. b. Prove that the sequence $(p_n)$ is decreasing. c. Is the sequence $(p_n)$ convergent?

Part A: establishing an inequality
On the interval $[ 0 ; + \infty [$, we define the function $f$ by $f ( x ) = x - \ln ( x + 1 )$.

1. Study the monotonicity of the function $f$ on the interval $[ 0 ; + \infty [$.
2. Deduce that for all $x \in [ 0 ; + \infty [ , \ln ( x + 1 ) \leqslant x$.

Part B: application to the study of a sequence
We set $u _ { 0 } = 1$ and for every natural number $n , u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right)$. We admit that the sequence with general term $u _ { n }$ is well defined.

1. Calculate an approximate value to $10 ^ { - 3 }$ of $u _ { 2 }$.
2. a. Prove by induction that for every natural number $n , \quad u _ { n } \geqslant 0$. b. Prove that the sequence $( u _ { n } )$ is decreasing, and deduce that for every natural number $n , \quad u _ { n } \leqslant 1$. c. Show that the sequence $( u _ { n } )$ is convergent.
3. Let $\ell$ denote the limit of the sequence $( u _ { n } )$ and we admit that $\ell = f ( \ell )$, where $f$ is the function defined in Part A. Deduce the value of $\ell$.
4. a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $( u _ { n } )$ are less than $10 ^ { - p }$. b. Determine the smallest natural number $n$ from which all terms of the sequence $( u _ { n } )$ are less than $10 ^ { - 15 }$.

We consider the sequence $\left(u_n\right)$ defined for every integer $n \geqslant 0$ by: $\left\{\begin{array}{l}u_{n+1} = 3 - \dfrac{10}{u_n + 4}\\u_0 = 5\end{array}\right.$

Part A:

1. Determine the exact value of $u_1$ and $u_2$.
2. Prove by induction that for every natural integer $n$, $u_n \geqslant 1$.
3. Prove that, for every natural integer $n$, $u_{n+1} - u_n = \dfrac{\left(1 - u_n\right)\left(u_n + 2\right)}{u_n + 4}$.
4. Deduce the direction of variation of the sequence $(u_n)$.
5. Justify that the sequence $\left(u_n\right)$ converges.

Part B:

We consider the sequence $(v_n)$ defined for every natural integer $n$ by $v_n = \dfrac{u_n - 1}{u_n + 2}$.

1. a. Prove that $\left(v_n\right)$ is a geometric sequence whose common ratio and first term $v_0$ we will determine. b. Express $v_n$ as a function of $n$.
   Deduce that for every natural integer $n$, $v_n \neq 1$.
2. Prove that for every natural integer $n$, $u_n = \dfrac{2v_n + 1}{1 - v_n}$.
3. Deduce the limit of the sequence $(u_n)$.

Part C:

We consider the algorithm below.

$u \leftarrow 5$

$n \leftarrow 0$

While $u \geqslant 1.01$

$n \leftarrow n + 1$

$u \leftarrow 3 - \dfrac{10}{u + 4}$

End While

1. After execution of the algorithm, what value is contained in the variable $n$?
2. Using parts A and B, interpret this value.

Part A: establishing an inequality

On the interval $[0; +\infty[$, we define the function $f$ by $f(x) = x - \ln(x+1)$.

1. Study the monotonicity of the function $f$ on the interval $[0; +\infty[$.
2. Deduce that for all $x \in [0; +\infty[,\; \ln(x+1) \leqslant x$.

Part B: application to the study of a sequence

We set $u_0 = 1$ and for all natural number $n$, $u_{n+1} = u_n - \ln(1 + u_n)$. We admit that the sequence with general term $u_n$ is well defined.

1. Calculate an approximate value to $10^{-3}$ of $u_2$.
2. a. Prove by induction that for all natural number $n$, $u_n \geqslant 0$. b. Prove that the sequence $(u_n)$ is decreasing, and deduce that for all natural number $n$, $u_n \leqslant 1$. c. Show that the sequence $(u_n)$ is convergent.
3. Let $\ell$ denote the limit of the sequence $(u_n)$ and we admit that $\ell = f(\ell)$, where $f$ is the function defined in part A. Deduce the value of $\ell$.
4. a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $(u_n)$ are less than $10^{-p}$. b. Determine the smallest natural number $n$ from which all terms of the sequence $(u_n)$ are less than $10^{-15}$.

For candidates who have not followed the specialization course
Let $f$ be the function defined on the interval $[ 0 ; 4]$ by $$f ( x ) = \frac { 2 + 3 x } { 4 + x }$$

Part A

We consider the sequence ( $u _ { n }$ ) defined by: $$u _ { 0 } = 3 \text { and for all natural integer } n , u _ { n + 1 } = f \left( u _ { n } \right) .$$ It is admitted that this sequence is well defined.

1. Calculate $u _ { 1 }$.
2. Show that the function $f$ is increasing on the interval $[ 0 ; 4 ]$.
3. Show that for all natural integer $n$, $$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 3$$
4. a. Show that the sequence ( $u _ { n }$ ) is convergent. b. We call $\ell$ the limit of the sequence ( $u _ { n }$ ); show the equality: $$\ell = \frac { 2 + 3 \ell } { 4 + \ell }$$ c. Determine the value of the limit $\ell$.

Part B

We consider the sequence $\left( v _ { n } \right)$ defined by: $$v _ { 0 } = 0.1 \text { and for all natural integer } n , v _ { n + 1 } = f \left( v _ { n } \right) .$$

1. We give in the Annex the representative curve $\mathscr { C } _ { f }$ of the function $f$ and the line $D$ with equation $y = x$. Place on the $x$-axis by geometric construction the terms $v _ { 1 } , v _ { 2 }$ and $v _ { 3 }$ on the annex, to be returned with the copy. What conjecture can be formulated about the direction of variation and the behavior of the sequence ( $v _ { n }$ ) as $n$ tends to infinity?
2. a. Show that for all natural integer $n$, $$1 - v _ { n + 1 } = \left( \frac { 2 } { 4 + v _ { n } } \right) \left( 1 - v _ { n } \right)$$ b. Show by induction that for all natural integer $n$, $$0 \leqslant 1 - v _ { n } \leqslant \left( \frac { 1 } { 2 } \right) ^ { n }$$
3. Does the sequence $\left( v _ { n } \right)$ converge? If so, specify its limit.

Let $f$ be the function defined on the interval $] - \frac { 1 } { 3 } ; + \infty [$ by: $$f ( x ) = \frac { 4 x } { 1 + 3 x }$$ We consider the sequence $(u _ { n })$ defined by: $u _ { 0 } = \frac { 1 } { 2 }$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$.

1. Calculate $u _ { 1 }$.
2. We admit that the function $f$ is increasing on the interval $] - \frac { 1 } { 3 } ; + \infty [$. a. Show by induction that, for every natural number $n$, we have: $\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 2$. b. Deduce that the sequence $(u _ { n })$ is convergent. c. We call $\ell$ the limit of the sequence $(u _ { n })$. Determine the value of $\ell$.
3. a. Copy and complete the Python function below which, for every positive real number $E$, determines the smallest value $P$ such that: $1 - u _ { P } < E$. \begin{verbatim} def seuil(E) : u=0.5 n = 0 while u = n = n + 1 return n \end{verbatim} b. Give the value returned by this program in the case where $E = 10 ^ { - 4 }$.
4. We consider the sequence $(v _ { n })$ defined, for every natural number $n$, by: $$v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$$ a. Show that the sequence $(v _ { n })$ is geometric with common ratio 4. Deduce, for every natural number $n$, the expression of $v _ { n }$ as a function of $n$. b. Prove that, for every natural number $n$, we have: $u _ { n } = \frac { v _ { n } } { v _ { n } + 1 }$. c. Show then that, for every natural number $n$, we have: $$u _ { n } = \frac { 1 } { 1 + 0.25 ^ { n } }$$ Find by calculation the limit of the sequence $(u _ { n })$.

We consider the sequences $(u_{n})$ and $(v_{n})$ defined for every natural integer $n$ by:
$$\left\{ \begin{array}{l} u_{0} = v_{0} = 1 \\ u_{n+1} = u_{n} + v_{n} \\ v_{n+1} = 2u_{n} + v_{n} \end{array} \right.$$
Throughout the rest of the exercise, we assume that the sequences $(u_{n})$ and $(v_{n})$ are strictly positive.

1. a. Calculate $u_{1}$ and $v_{1}$. b. Prove that the sequence $(v_{n})$ is strictly increasing, then deduce that for every natural integer $n$, $v_{n} \geqslant 1$. c. Prove by induction that for every natural integer $n$, we have: $u_{n} \geqslant n + 1$. d. Deduce the limit of the sequence $(u_{n})$.
2. We set, for every natural integer $n$: $$r_{n} = \frac{v_{n}}{u_{n}}.$$ We assume that: $$r_{n}^{2} = 2 + \frac{(-1)^{n+1}}{u_{n}^{2}}$$ a. Prove that for every natural integer $n$: $$-\frac{1}{u_{n}^{2}} \leqslant \frac{(-1)^{n+1}}{u_{n}^{2}} \leqslant \frac{1}{u_{n}^{2}}.$$ b. Deduce: $$\lim_{n \rightarrow +\infty} \frac{(-1)^{n+1}}{u_{n}^{2}}$$ c. Determine the limit of the sequence $\left(r_{n}^{2}\right)$ and deduce that $\left(r_{n}\right)$ converges to $\sqrt{2}$. d. Prove that for every natural integer $n$, $$r_{n+1} = \frac{2 + r_{n}}{1 + r_{n}}$$ e. Consider the following program written in Python language: \begin{verbatim} def seuil() : n = 0 r = l while abs(r-sqrt(2)) > 10**(-4) : r = (2+r)/(1+r) n = n+1 return n \end{verbatim} (abs denotes absolute value, sqrt the square root and $10^{**}(-4)$ represents $10^{-4}$). The value of $n$ returned by this program is 5. What does it correspond to?

Part A

Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by $$f ( x ) = \frac { \ln x } { x }$$ where ln denotes the natural logarithm function.

1. Give the limit of the function $f$ at $+ \infty$.
2. We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function. a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
   

   $x$	1		e	$+ \infty$
   $f ^ { \prime } ( x )$		+	0	-

   
   c. Draw up the complete variation table of the function $f$.
3. Let $k$ be a non-negative real number. a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$. b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.

Part B

Let $g$ be the function defined on $\mathbb { R }$ by: $$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$ We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.

1. Justify that the function $g$ is increasing on $\mathbb { R }$.
2. Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
3. Deduce that the sequence $( u _ { n } )$ is convergent.

We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation: $$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$

1. Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
2. Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.

We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, $$u_{n+1} = \frac{1}{2}\left(u_n + \frac{11}{u_n}\right)$$ We admit that the sequence $(u_n)$ is well defined.
Part A - Study of sequence $(u_n)$

1. Give $u_1$ and $u_2$ in the form of irreducible fractions.
2. We consider the function $f$ defined on the interval $]0; +\infty[$ by: $$f(x) = \frac{1}{2}\left(x + \frac{11}{x}\right)$$ Prove that function $f$ is increasing on the interval $[\sqrt{11}; +\infty[$.
3. Prove by induction that for every natural number $n$, we have: $u_n \geqslant u_{n+1} \geqslant \sqrt{11}$.
4. Deduce that the sequence $(u_n)$ converges to a real limit. We denote this limit by $a$.
5. After determining and solving an equation of which $a$ is a solution, specify the exact value of $a$.

Part B - Geometric application
For every natural number $n$, we consider a rectangle $R_n$ with area 11 whose width is denoted $\ell_n$ and length $L_n$. The sequence $(L_n)$ is defined by $L_0 = 5$ and, for every natural number $n$, $$L_{n+1} = \frac{L_n + \ell_n}{2}$$

1. a. Explain why $\ell_0 = 2.2$. b. Establish that for every natural number $n$, $$\ell_n = \frac{11}{L_n}.$$
2. Verify that the sequence $(L_n)$ corresponds to the sequence $(u_n)$ from Part A.
3. Show that for every natural number $n$, we have $\ell_n \leqslant \sqrt{11} \leqslant L_n$.
4. We admit that the sequences $(L_n)$ and $(\ell_n)$ both converge to $\sqrt{11}$. Interpret this result geometrically in the context of Part B.
5. Here is a script, written in Python language, relating to the sequences studied in this part: \begin{verbatim} def heron(n): L=5 l=2.2 for i in range(n): L = (L+l) / 2 l = 11 / L return round(l, 6), round(L, 6) \end{verbatim} We recall that the Python function round$(\mathrm{x}, \mathrm{k})$ returns a rounded version of the number x with k decimal places. a. If the user types heron(3) in a Python execution console, what output values does he obtain for $\ell$ and $L$? b. Give an interpretation of these two values.

We consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 400$ and for every natural integer $n$:
$$u _ { n + 1 } = 0,9 u _ { n } + 60 .$$

1. a. Calculate $u _ { 1 }$ and $u _ { 2 }$. b. Conjecture the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$
2. Show, by induction, that for every natural integer $n$, we have the inequality $$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 600$$
3. a. Show that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. b. Determine the limit of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify.
4. A function is given written in Python language: \begin{verbatim} def mystere(seuil) : n=0 u=400 while u <= seuil : n = n+1 u=0.9*u+60 return n \end{verbatim} What value do we obtain by typing in the Python console: mystere(500)?

Part B: A fruit grower owns an orchard where he has room to grow a maximum of 500 trees. Each year he sells $10\%$ of the trees in his orchard and then he plants 60 new trees. The orchard has 400 trees in 2023. The fruit grower thinks he will be able to continue selling and planting trees at the same rate in the coming years. Will he face a space problem in his orchard? Explain your answer.

Part A

Consider the function $f$ defined by :
$$f ( x ) = x - \ln ( 1 + x ) .$$

1. Justify that the function $f$ is defined on the interval $] - 1 ; + \infty [$.
2. We admit that the function $f$ is differentiable on $] - 1 ; + \infty [$.

Determine the expression of its derivative function $f ^ { \prime } ( x )$.
3. a. Deduce the direction of variation of the function $f$ on the interval $] - 1 ; + \infty [$. b. Deduce the sign of the function $f$ on the interval $] - 1 ; 0 [$.
4. a. Show that, for all $x$ belonging to the interval $] - 1 ; + \infty [$, we have :
$$f ( x ) = \ln \left( \frac { \mathrm { e } ^ { x } } { 1 + x } \right) .$$
b. Deduce the limit at $+ \infty$ of the function $f$.

Part B

Consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 10$ and, for all natural integer $n$,
$$u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right) .$$
We admit that the sequence ( $u _ { n }$ ) is well defined.

1. Give the value rounded to the nearest thousandth of $u _ { 1 }$.
2. Using question 3. a. of Part A , prove by induction that, for all natural integer $n$, we have $u _ { n } \geqslant 0$.
3. Prove that the sequence ( $u _ { n }$ ) is decreasing.
4. Deduce from the previous questions that the sequence ( $u _ { n }$ ) converges.
5. Determine the limit of the sequence $\left( u _ { n } \right)$.

Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \frac{3}{4}x^2 - 2x + 3$$

1. Draw the table of variations of $f$ on $\mathbb{R}$.
2. Deduce that for all $x$ belonging to the interval $\left[\frac{4}{3}; 2\right]$, $f(x)$ belongs to the interval $\left[\frac{4}{3}; 2\right]$.
3. Prove that for all real $x$, $x \leq f(x)$. For this, one may prove that for all real $x$: $$f(x) - x = \frac{3}{4}(x - 2)^2.$$

Consider the sequence $(u_n)$ defined by a real $u_0$ and for all natural integer $n$: $$u_{n+1} = f(u_n).$$ We have therefore, for all natural integer $n$, $$u_{n+1} = \frac{3}{4}u_n^2 - 2u_n + 3.$$

1. Study of the case: $\frac{4}{3} \leq u_0 \leq 2$. a. Prove by induction that, for all natural integer $n$, $$u_n \leq u_{n+1} \leq 2.$$ b. Deduce that the sequence $(u_n)$ is convergent. c. Prove that the limit of the sequence is equal to 2.
2. Study of the particular case: $u_0 = 3$. It is admitted that in this case the sequence $(u_n)$ tends to $+\infty$. Copy and complete the following ``threshold'' function written in Python, so that it returns the smallest value of $n$ such that $u_n$ is greater than or equal to 100. \begin{verbatim} def seuil() : u = 3 n = 0 while ... u = ... n = ... return n \end{verbatim}
3. Study of the case: $u_0 > 2$. Using a proof by contradiction, show that $(u_n)$ is not convergent.

We consider the sequence $(u_n)$ defined by $u_0 = 8$ and, for every natural number $n$, $$u_{n+1} = \frac{6u_n + 2}{u_n + 5}.$$

1. Calculate $u_1$.
2. Let $f$ be the function defined on the interval $[0; +\infty[$ by: $$f(x) = \frac{6x + 2}{x + 5}.$$ Thus, for every natural number $n$, we have: $u_{n+1} = f(u_n)$.
   a. Prove that the function $f$ is strictly increasing on the interval $[0; +\infty[$. Deduce that for every real number $x > 2$, we have $f(x) > 2$.
   b. Prove by induction that, for every natural number $n$, we have $u_n > 2$.
   
3. We admit that, for every natural number $n$, we have: $$u_{n+1} - u_n = \frac{(2 - u_n)(u_n + 1)}{u_n + 5}.$$
   a. Prove that the sequence $(u_n)$ is decreasing.
   b. Deduce that the sequence $(u_n)$ is convergent.
   
4. We define the sequence $(v_n)$ for every natural number by: $$v_n = \frac{u_n - 2}{u_n + 1}.$$
   a. Calculate $v_0$.
   b. Prove that $(v_n)$ is a geometric sequence with common ratio $\dfrac{4}{7}$.
   c. Determine, by justifying, the limit of $(v_n)$. Deduce the limit of $(u_n)$.
   
5. We consider the Python function threshold below, where A is a real number strictly greater than 2. \begin{verbatim} def seuil(A): n = 0 u = 8 while u > A: u = (6*u + 2)/(u + 5) n = n + 1 return n \end{verbatim} Give, without justification, the value returned by the command \texttt{seuil(2.001)} then interpret this value in the context of the exercise.

We consider the function $g$ defined on the interval $[0; 1]$ by $$g(x) = 2x - x^2$$

1. Show that the function $g$ is strictly increasing on the interval $[0; 1]$ and specify the values of $g(0)$ and $g(1)$.

We consider the sequence $\left(u_n\right)$ defined by $\left\{\begin{array}{l} u_0 = \dfrac{1}{2} \\ u_{n+1} = g\left(u_n\right) \end{array}\right.$ for every natural number $n$.

1. Calculate $u_1$ and $u_2$.
2. Prove by induction that, for every natural number $n$, we have: $0 < u_n < u_{n+1} < 1$.
3. Deduce that the sequence $(u_n)$ is convergent.
4. Determine the limit $\ell$ of the sequence $(u_n)$.

We consider the sequence $(\nu_n)$ defined for every natural number $n$ by $\nu_n = \ln\left(1 - u_n\right)$.

1. Prove that the sequence $\left(v_n\right)$ is a geometric sequence with common ratio 2 and specify its first term.
2. Deduce an expression for $v_n$ as a function of $n$.
3. Deduce an expression for $u_n$ as a function of $n$ and find again the limit determined in question 5.
4. Copy and complete the Python script below so that it returns the rank $n$ from which the sequence exceeds 0.95. \begin{verbatim} def seuil() : n=0 u=0.5 while u < 0.95: n=... u=... return n \end{verbatim}

Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by $$f ( x ) = 2 x \mathrm { e } ^ { - x } .$$ It is admitted that the function $f$ is differentiable on the interval $[ 0 ; 1 ]$.

1. 1. [a.] Solve on the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.
   2. [b.] Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = 2 ( 1 - x ) \mathrm { e } ^ { - x } .$$
   3. [c.] Give the table of variations of the function $f$ on the interval $[ 0 ; 1 ]$.
   
   Consider the sequence $(u_n)$ defined by $u_0 = 0,1$ and for all natural integer $n$, $$u_{n+1} = f(u_n).$$
   
2. 1. [a.] Prove by induction that, for all natural integer $n$, $$0 \leqslant u_n < u_{n+1} \leqslant 1.$$
   2. [b.] Deduce that the sequence $(u_n)$ is convergent.
3. Prove that the limit of the sequence $(u_n)$ is $\ln(2)$.
4. 1. [a.] Justify that for all natural integer $n$, $\ln(2) - u_n$ is positive.
   2. [b.] It is desired to write a Python script that returns an approximate value of $\ln(2)$ by default to within $10^{-4}$, as well as the number of steps to achieve this. Copy and complete the script below so that it answers the problem posed. \begin{verbatim} def seuil() : n = 0 u=0.1 while ln (2) - u ...0.0001 : n=n+1 u=... return (u,n) \end{verbatim}
   3. [c.] Give the value of the variable $n$ returned by the function seuil().

Part A
We consider the function $f$ defined on the interval $[0; +\infty[$ by $$f(x) = \sqrt{x+1}.$$ We admit that this function is differentiable on this same interval.

1. Prove that the function $f$ is increasing on the interval $[0; +\infty[$.
2. Prove that for every real number $x$ belonging to the interval $[0; +\infty[$: $$f(x) - x = \frac{-x^2 + x + 1}{\sqrt{x+1} + x}.$$
3. Deduce from this that on the interval $[0; +\infty[$ the equation $f(x) = x$ admits as unique solution: $$\ell = \frac{1+\sqrt{5}}{2}.$$

Part B
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, by $u_{n+1} = f(u_n)$ where $f$ is the function studied in part A. We admit that the sequence with general term $u_n$ is well defined for every natural number $n$.

1. Prove by induction that for every natural number $n$, we have $$1 \leqslant u_{n+1} \leqslant u_n.$$
2. Deduce from this that the sequence $(u_n)$ converges.
3. Prove that the sequence $(u_n)$ converges to $\ell = \frac{1+\sqrt{5}}{2}$.
4. We consider the Python script below: \begin{verbatim} from math import * def seuil(n): u=5 i=0 while abs(u-l)>=10**(-n): u=sqrt(u+1) i=i+1 return(i) \end{verbatim} We recall that the command $\mathbf{abs}(\mathbf{x})$ returns the absolute value of $x$.
   1. [a.] Give the value returned by \texttt{seuil(2)}.
   2. [b.] The value returned by \texttt{seuil(4)} is 9. Interpret this value in the context of the exercise.

We consider the numerical sequence $(u_n)$ defined by its first term $u_0 = 2$ and for every natural number $n$, by: $$u_{n+1} = \frac{2u_n + 1}{u_n + 2}$$ We admit that the sequence $(u_n)$ is well defined.

1. Calculate the term $u_1$.
2. We define the sequence $(a_n)$ for every natural number $n$, by: $$a_n = \frac{u_n}{u_n - 1}$$ We admit that the sequence $(a_n)$ is well defined. a. Calculate $a_0$ and $a_1$. b. Prove that, for every natural number $n$, $a_{n+1} = 3a_n - 1$. c. Prove by induction that, for every natural number $n$ greater than or equal to 1, $$a_n \geqslant 3n - 1$$ d. Deduce the limit of the sequence $(a_n)$.
3. We wish to study the limit of the sequence $(u_n)$. a. Prove that for every natural number $n$, $u_n = \frac{a_n}{a_n - 1}$. b. Deduce the limit of the sequence $(u_n)$.
4. We admit that the sequence $(u_n)$ is decreasing.
   We consider the following program written in Python: \begin{verbatim} def algo(p): u=2 n=0 while u-1>p: u=(2*u+1)/(u+2) n=n+1 return (n,u) \end{verbatim} a. Interpret the values $n$ and u returned by the call to the function algo(p) in the context of the exercise. b. Give, without justification, the value of $n$ for $p = 0.001$.

One of the objectives of this exercise is to determine an approximation of the real number $\ln ( 2 )$, using one of the methods of the English mathematician Henry Briggs in the XVI${}^{\text{th}}$ century.
We denote by ( $u _ { n }$ ) the sequence defined by:
$$u _ { 0 } = 2 \quad \text { and, for every natural number } n , \quad u _ { n + 1 } = \sqrt { u _ { n } }$$

Part A

1. a. Give the exact value of $u _ { 1 }$ and $u _ { 2 }$. b. Make a conjecture, using a calculator, about the direction of variation and the possible limit of the sequence.
2. a. Show by induction that for every natural number $n , \quad 1 \leqslant u _ { n + 1 } \leqslant u _ { n }$. b. Deduce that the sequence ( $u _ { n }$ ) is convergent. c. Solve in the interval [ $0 ; + \infty$ [ the equation $\sqrt { x } = x$. d. Determine, by justifying, the limit of the sequence $\left( u _ { n } \right)$.

Part B

We denote by ( $\nu _ { n }$ ) the sequence defined for every natural number $n$ by $\nu _ { n } = \ln \left( u _ { n } \right)$.

1. a. Prove that the sequence ( $v _ { n }$ ) is geometric with common ratio $\frac { 1 } { 2 }$. b. Express $v _ { n }$ as a function of $n$, for every natural number $n$. c. Deduce that, for every natural number $n , \quad \ln ( 2 ) = 2 ^ { n } \ln \left( u _ { n } \right)$.
2. We have traced below in an orthonormal coordinate system the curve $\mathscr { C }$ of the function ln and the tangent T to the curve $\mathscr { C }$ at the point with abscissa 1. An equation of the line T is $y = x - 1$. The points $\mathrm { A } _ { 0 } , \mathrm {~A} _ { 1 } , \mathrm {~A} _ { 2 }$ have abscissas $u _ { 0 } , u _ { 1 }$ and $u _ { 2 }$ respectively and ordinate 0. We decide to take $x - 1$ as an approximation of $\ln ( x )$ when $x$ belongs to the interval $] 0,99 ; 1,01 [$. a. Using a calculator, determine the smallest natural number $k$ such that $u _ { k }$ belongs to the interval $] 0,99 ; 1,01 [$ and give an approximate value of $u _ { k }$ to $10 ^ { - 5 }$ near. b. Deduce an approximation of $\ln \left( u _ { k } \right)$. c. Deduce from questions 1.c. and 2.b. of Part B an approximation of $\ln ( 2 )$.
3. We generalize the previous method to any real number $a$ strictly greater than 1.
   Copy and complete the algorithm below so that the call Briggs(a) returns an approximation of $\ln ( a )$.
   We recall that the instruction in Python language sqrt (a) corresponds to $\sqrt { a }$.
   \begin{verbatim} from math import* def Briggs(a): n = 0 while a >= 1.01: a = sqrt(a) n = n+1 L =... return L \end{verbatim}

Consider the function $f$ defined for all real $x$ by: $$f ( x ) = \ln \left( \mathrm { e } ^ { \frac { x } { 2 } } + 2 \right)$$ It is admitted that the function $f$ is differentiable on $\mathbb { R }$. Consider the sequence $(u_n)$ defined by $u _ { 0 } = \ln ( 9 )$ and, for all natural integer $n$, $$u _ { n + 1 } = f \left( u _ { n } \right)$$

1. Show that the function $f$ is strictly increasing on $\mathbb { R }$.
2. Show that $f ( 2 \ln ( 2 ) ) = 2 \ln ( 2 )$.
3. Show that $u _ { 1 } = \ln ( 5 )$.
4. Show by induction that for all natural integer $n$, we have: $$2 \ln ( 2 ) \leqslant u _ { n + 1 } \leqslant u _ { n }$$
5. Deduce that the sequence $(u_n)$ converges.
6. a. Solve in $\mathbb { R }$ the equation $X ^ { 2 } - X - 2 = 0$. b. Deduce the set of solutions on $\mathbb { R }$ of the equation: $$\mathrm { e } ^ { x } - \mathrm { e } ^ { \frac { x } { 2 } } - 2 = 0$$ c. Deduce the set of solutions on $\mathbb { R }$ of the equation $f ( x ) = x$. d. Determine the limit of the sequence $\left( u _ { n } \right)$.

Exercise 3
The purpose of this exercise is to study the convergence of two sequences towards the same limit.
Part A
Consider the function $f$ defined on $[2;+\infty[$ by $$f(x) = \sqrt{3x-2}$$

1. Justify the elements of the variation table below:
   

   $x$	2	$+\infty$
   		$+\infty$
   $f(x)$
   	2

   
   We admit that the sequence $(u_n)$ satisfying $u_0 = 6$ and, for all natural number $n$, $u_{n+1} = f(u_n)$ is well defined.
   
2. 1. [a.] Prove by induction that, for all natural number $n$: $2 \leqslant u_{n+1} \leqslant u_n \leqslant 6$.
   2. [b.] Deduce that the sequence $(u_n)$ converges.
3. We call $\ell$ the limit of $(u_n)$.
   We admit that it is a solution of the equation $f(x) = x$. Determine the value of $\ell$.
   
4. Consider the rank function written below in Python language.
   We recall that $\operatorname{sqrt}(x)$ returns the square root of the number $x$.
   \begin{verbatim} from math import * def rang(a) : u=6 n=0 while u >= a : u = sqrt(3*u - 2) n = n+1 return n \end{verbatim}
   
   1. [a.] Why can we affirm that rang(2.000001) returns a value?
   2. [b.] For which values of the parameter $a$ does the instruction rang($a$) return a result?

Part B
We admit that the sequence $(v_n)$ satisfying $v_0 = 6$ and, for all natural number $n$, $v_{n+1} = 3 - \dfrac{2}{v_n}$ is well defined.

1. Calculate $v_1$.
2. For all natural number $n$, we admit that $v_n \neq 2$ and we set: $$w_n = \frac{v_n - 1}{v_n - 2}$$
   1. [a.] Prove that the sequence $(w_n)$ is geometric with ratio 2 and specify its first term $w_0$.
   2. [b.] We admit that, for all natural number $n$, $$w_n - 1 = \frac{1}{v_n - 2}$$ Deduce that, for all natural number $n$, $$v_n = 2 + \frac{1}{1{,}25 \times 2^n - 1}$$
   3. [c.] Calculate the limit of $(v_n)$.
3. Determine the smallest natural number $n$ for which $v_n < 2{,}01$ by solving the inequality.

Part C
Using the previous parts, determine the smallest integer $N$ such that for all $n \geqslant N$, the terms $v_n$ and $u_n$ belong to the interval $]1{,}99;2{,}01[$.

Let $A$ be a $n \times m$ matrix with real entries, and let $B = A A ^ { t }$ and let $\alpha$ be the supremum of $x ^ { t } B x$ where supremum is taken over all vectors $x \in \mathbb { R } ^ { n }$ with norm less than or equal to 1. Consider $$C _ { k } = I + \sum _ { j = 1 } ^ { k } B ^ { j }$$ Show that the sequence of matrices $C _ { k }$ converges if and only if $\alpha < 1$.

Show that $$\lim _ { x \rightarrow \infty } \frac { x ^ { 100 } \ln ( x ) } { e ^ { x } \tan ^ { - 1 } \left( \frac { \pi } { 3 } + \sin x \right) } = 0$$

Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that
$$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$