If the series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges and $a _ { n } > 0$ for all $n$, which of the following must be true?
(A) $\lim _ { n \rightarrow \infty } \left| \frac { a _ { n + 1 } } { a _ { n } } \right| = 0$
(B) $\left| a _ { n } \right| < 1$ for all $n$
(C) $\sum _ { n = 1 } ^ { \infty } a _ { n } = 0$
(D) $\sum _ { n = 1 } ^ { \infty } n a _ { n }$ diverges.
(E) $\sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { n }$ converges.

Question 2: Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by $v_n = \frac{3n}{n+2}$. We seek to determine the limit of $v_n$ as $n$ tends to $+\infty$.

a. $\lim_{n\rightarrow+\infty} v_n = 1$	b. $\lim_{n\rightarrow+\infty} v_n = 3$	c. $\lim_{n\rightarrow+\infty} v_n = \frac{3}{2}$	\begin{tabular}{l} d. We cannot
determine it

\hline \end{tabular}

Exercise 4 (7 points) — Main topics covered: sequences, functions, antiderivatives.
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. Consider the sequence $(u_n)$ defined for all natural number $n$ by $$u_n = \frac{(-1)^n}{n+1}.$$ We can affirm that: a. the sequence $(u_n)$ diverges to $+\infty$. b. the sequence $(u_n)$ diverges to $-\infty$. c. the sequence $(u_n)$ has no limit. d. the sequence $(u_n)$ converges.
   In questions 2 and 3, we consider two sequences $(v_n)$ and $(w_n)$ satisfying the relation: $$w_n = \mathrm{e}^{-2v_n} + 2.$$
   
2. Let $a$ be a strictly positive real number. We have $v_0 = \ln(a)$. a. $w_0 = \dfrac{1}{a^2} + 2$ b. $w_0 = \dfrac{1}{a^2 + 2}$ c. $w_0 = -2a + 2$ d. $w_0 = \dfrac{1}{-2a} + 2$
   
3. We know that the sequence $(v_n)$ is increasing. We can affirm that the sequence $(w_n)$ is: a. decreasing and bounded above by 3. b. decreasing and bounded below by 2. c. increasing and bounded above by 3. d. increasing and bounded below by 2.
   
4. Consider the sequence $(a_n)$ defined as follows: $$a_0 = 2 \text{ and, for all natural number } n, \quad a_{n+1} = \frac{1}{3}a_n + \frac{8}{3}.$$ For all natural number $n$, we have: a. $a_n = 4 \times \left(\dfrac{1}{3}\right)^n - 2$ b. $a_n = -\dfrac{2}{3^n} + 4$ c. $a_n = 4 - \left(\dfrac{1}{3}\right)^n$ d. $a_n = 2 \times \left(\dfrac{1}{3}\right)^n + \dfrac{8n}{3}$
   
5. Consider a sequence $(b_n)$ such that, for all natural number $n$, we have: $$b_{n+1} = b_n + \ln\left(\frac{2}{(b_n)^2 + 3}\right)$$ We can affirm that: a. the sequence $(b_n)$ is increasing. b. the sequence $(b_n)$ is decreasing. c. the sequence $(b_n)$ is not monotone. d. the direction of variation of the sequence $(b_n)$ depends on $b_0$.
   
6. Consider the function $g$ defined on the interval $]0; +\infty[$ by: $$g(x) = \frac{\mathrm{e}^x}{x}$$ We denote $\mathcal{C}_g$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathcal{C}_g$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
   
7. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2+1}$$ Let $F$ be an antiderivative on $\mathbb{R}$ of the function $f$. For all real $x$, we have: a. $F(x) = \dfrac{1}{2}x^2\mathrm{e}^{x^2+1}$ b. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2+1}$ c. $F(x) = \mathrm{e}^{x^2+1}$ d. $F(x) = \dfrac{1}{2}\mathrm{e}^{x^2+1}$

Exercise 3 — Theme: Functions; Sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. Let $g$ be the function defined on $\mathbb{R}$ by $g(x) = x^{1000} + x$. We can affirm that: a. the function $g$ is concave on $\mathbb{R}$. b. the function $g$ is convex on $\mathbb{R}$. c. the function $g$ has exactly one inflection point. d. the function $g$ has exactly two inflection points.
2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Let $f'$ denote its derivative function. Let $\mathscr{C}$ denote the representative curve of $f$. Let $\Gamma$ denote the representative curve of $f'$. The curve $\Gamma$ is plotted below. Let $T$ denote the tangent to the curve $\mathscr{C}$ at the point with abscissa 0. We can affirm that the tangent $T$ is parallel to the line with equation: a. $y = x$ b. $y = 0$ c. $y = 1$ d. $x = 0$
3. Consider the sequence $(u_n)$ defined for every natural number $n$ by $u_n = \frac{(-1)^n}{n+1}$. We can affirm that the sequence $(u_n)$ is: a. bounded above and not bounded below. b. bounded below and not bounded above. c. bounded. d. not bounded above and not bounded below.
4. Let $k$ be a non-zero real number. Let $(v_n)$ be a sequence defined for every natural number $n$. Suppose that $v_0 = k$ and that for all $n$, we have $v_n \times v_{n+1} < 0$. We can affirm that $v_{10}$ is: a. positive. b. negative. c. of the same sign as $k$. d. of the same sign as $-k$.
5. Consider the sequence $(w_n)$ defined for every natural number $n$ by: $$w_{n+1} = 2w_n - 4 \quad \text{and} \quad w_2 = 8.$$ We can affirm that: a. $w_0 = 0$ b. $w_0 = 5$. c. $w_0 = 10$. d. It is not possible to calculate $w_0$.
6. Consider the sequence $(a_n)$ defined for every natural number $n$ by: $$a_{n+1} = \frac{\mathrm{e}^n}{\mathrm{e}^n + 1} a_n \quad \text{and} \quad a_0 = 1.$$ We can affirm that: a. the sequence $(a_n)$ is strictly increasing. b. the sequence $(a_n)$ is strictly decreasing. c. the sequence $(a_n)$ is not monotone. d. the sequence $(a_n)$ is constant.
7. A cell reproduces by dividing into two identical cells, which divide in turn, and so on. The generation time is defined as the time required for a given cell to divide into two cells. 1 cell was placed in culture. After 4 hours, there are approximately 4000 cells. We can affirm that the generation time is approximately equal to: a. less than one minute. b. 12 minutes. c. 20 minutes. d. 1 hour.

Consider a sequence $( u _ { n } )$ such that, for every natural integer, we have: $$1 + \left( \frac { 1 } { 4 } \right) ^ { n } \leqslant u _ { n } \leqslant 2 - \frac { n } { n + 1 }$$ We can affirm that the sequence $\left( u _ { n } \right)$: a. converges to $2$; b. converges to $1$; c. diverges to $+ \infty$; d. has no limit.

Exercise 2 — 7 points

Topics: Sequences, Functions This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points. To answer, indicate on your answer sheet the question number and the letter of the chosen answer. No justification is required.

1. We consider the sequences $\left( a _ { n } \right)$ and $\left( b _ { n } \right)$ defined by $a _ { 0 } = 1$ and, for every natural number $n$, $a _ { n + 1 } = 0.5 a _ { n } + 1$ and $b _ { n } = a _ { n } - 2$. We can affirm that: a. $\left( a _ { n } \right)$ is arithmetic; b. $\left( b _ { n } \right)$ is geometric; c. $\left( a _ { n } \right)$ is geometric; d. $\left( b _ { n } \right)$ is arithmetic.
2. In questions 2. and 3., we consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We can affirm that: a. $\left\{ \begin{array} { l } u _ { 2 } = 5 \\ v _ { 2 } = 3 \end{array} \right.$ b. $u _ { 2 } ^ { 2 } - 3 v _ { 2 } ^ { 2 } = - 2 ^ { 2 }$ c. $\frac { u _ { 2 } } { v _ { 2 } } = 1.75$ d. $5 u _ { 1 } = 3 v _ { 1 }$.
3. We consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We consider the program below written in Python language: \begin{verbatim} def valeurs() : u = 2 v = 1 for k in range(1,11) c = u u = u + 3*v v = c + v return (u, v) \end{verbatim} This program returns: a. $u _ { 11 }$ and $v _ { 11 }$; b. $u _ { 10 }$ and $v _ { 11 }$; c. the values of $u _ { n }$ and $v _ { n }$ for $n$ ranging from 1 to 10; d. $u _ { 10 }$ and $v _ { 10 }$.
4. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. The function $f$ is: a. concave on $[-2; 1]$; b. convex on $[-4; 0]$; c. convex on $[ - 2 ; 1 ]$; d. convex on $[ 0 ; 2 ]$.
5. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. We admit that the line (BC) is tangent to the curve $\mathscr { C } ^ { \prime }$ at point B. We have: a. $f ^ { \prime } ( 1 ) < 0$; b. $f ^ { \prime } ( 1 ) = 5$; c. $f ^ { \prime \prime } ( 1 ) > 0$; d. $f ^ { \prime \prime } ( 1 ) = - 5$.
6. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { x }$. The antiderivative $F$ of $f$ on $\mathbb { R }$ such that $F ( 0 ) = 1$ is defined by: a. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x }$; b. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x } - 2$; c. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x } + 1$; d. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x }$.

Consider the numerical sequence $(u_n)$ defined for all natural integer $n$ by
$$u_n = \frac{1 + 2^n}{3 + 5^n}$$
This sequence: a. diverges to $+\infty$ b. converges to $\frac{2}{5}$ c. converges to 0 d. converges to $\frac{1}{3}$.

For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
The following Python function, whose line 4 is incomplete, must return the smallest value of the integer $n$ such that $u_n > 10000$. \begin{verbatim} def seuil() : n=0 u=15 while ......: n=n+1 u=1,2*u+12 return(n) \end{verbatim} On line 4, we complete with: a. $\mathrm{u} \leqslant 10000$; b. $\mathrm{u} = 10000$ c. $\mathrm{u} > 10000$; d. $n \leqslant 10000$.

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 3$ and, for every natural number $n$,
$$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + \frac { 1 } { 2 } n + 1 .$$

Part A

This part is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.

1. The value of $u _ { 2 }$ is equal to: a. $\frac { 11 } { 4 }$ b. $\frac { 13 } { 2 }$ b. 2.7 c. 3.5
2. The sequence $\left( v _ { n } \right)$ defined, for every natural number $n$, by $v _ { n } = u _ { n } - n$ is: a. arithmetic with common difference $\frac { 1 } { 2 }$ b. geometric with common ratio $\frac { 1 } { 2 }$ c. constant. d. neither arithmetic nor geometric.
3. We consider the function below, written incompletely in Python language. $n$ denotes a non-zero natural number. We recall that in Python language ``i in range (n)'' means that i varies from 0 to $n - 1$.
   

   1	def terme $( \mathrm { n } )$
   2	$\mathrm { U } = 3$
   3	for i in range(n) :
   4	$\ldots \ldots \ldots \ldots \ldots \ldots \ldots$
   5	return U

   
   For terme(n) to return the value of $u _ { n }$, we can complete line 4 by: a. $\mathrm { U } = \mathrm { U } / 2 + ( \mathrm { i } + 1 ) / 2 + 1$ b. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { n } / 2 + 1$ c. $U = U / 2 + ( i - 1 ) / 2 + 1$ d. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { i } / 2 + 1$

Part B

1. Prove by induction that for every natural number $n$: $$n \leqslant u _ { n } \leqslant n + 3 .$$
2. Deduce the limit of the sequence $( u _ { n } )$.
3. Determine the limit of the sequence $\left( \frac { u _ { n } } { n } \right)$.

Let $x_n = \left(1 - \frac{1}{n}\right) \sin \frac{n\pi}{3}$, $n \geq 1$. Write $l = \liminf x_n$ and $s = \limsup x_n$. Choose the correct statement(s) from below:
(A) $-\frac{\sqrt{3}}{2} \leq l < s \leq \frac{\sqrt{3}}{2}$;
(B) $-\frac{1}{2} \leq l < s \leq \frac{1}{2}$;
(C) $l = -1$ and $s = 1$;
(D) $l = s = 0$.

What is the value of $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } + 6 n + 4 } - n \right)$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 3

When a sequence $\left\{ a _ { n } \right\}$ satisfies $n < a _ { n } < n + 1$ for all natural numbers $n$, what is the value of $\lim _ { n \rightarrow \infty } \frac { n ^ { 2 } } { a _ { 1 } + a _ { 2 } + \cdots + a _ { n } }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { n \rightarrow \infty } \frac { 3 + \left( \frac { 1 } { 3 } \right) ^ { n } } { 2 + \left( \frac { 1 } { 2 } \right) ^ { n } }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \sqrt { n ^ { 2 } + 2 n } - \sqrt { n ^ { 2 } + 1 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $\lim _ { n \rightarrow \infty } \frac { ( n + 1 ) ( 3 n - 1 ) } { 2 n ^ { 2 } + 1 }$? [2 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$

What is the value of $\lim _ { n \rightarrow \infty } \frac { 5 ^ { n + 1 } + 2 } { 5 ^ { n } + 3 ^ { n } }$? [2 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

What is the value of $\lim _ { n \rightarrow \infty } \frac { 5n ^ { 2 } + 1 } { 3n ^ { 2 } - 1 }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 4 } { 3 }$
(5) $\frac { 5 } { 3 }$

Find the value of $\lim _ { n \rightarrow \infty } \frac { 5 ^ { n } - 3 } { 5 ^ { n + 1 } }$. [2 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 }$
(5) 1

What is the value of $\lim _ { n \rightarrow \infty } \frac { 6 n ^ { 2 } - 3 } { 2 n ^ { 2 } + 5 n }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

What is the value of $\lim _ { n \rightarrow \infty } \frac { \sqrt { 9 n ^ { 2 } + 4 } } { 5 n - 2 }$? [2 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 4 } { 5 }$
(5) 1

What is the value of $\lim _ { n \rightarrow \infty } \frac { \frac { 5 } { n } + \frac { 3 } { n ^ { 2 } } } { \frac { 1 } { n } - \frac { 2 } { n ^ { 3 } } }$?

Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true?
(A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
(B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$
(C) $x_n \to 1$ for all $a > 0$
(D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{ a_n \}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.