We consider two sequences $(U _ { n })$ and $(V _ { n })$ defined on $\mathbb { N }$ such that:

• for every natural number $n$, $U _ { n } \leqslant V _ { n }$;
• $\lim _ { n \rightarrow + \infty } V _ { n } = 2$.

We can assert that: a. the sequence $(U _ { n })$ converges b. for every natural number $n$, $V _ { n } \leqslant 2$ c. the sequence $(U _ { n })$ diverges d. the sequence $(U _ { n })$ is bounded above

Exercise 4 — 5 points Theme: sequences, logarithm function, algorithms For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points:

1. Statement: The sequence $u$ defined for every natural integer $n$ by $u_{n} = \frac{(-1)^{n}}{n+1}$ is bounded.
2. Statement: Every bounded sequence is convergent.
3. Statement: Every increasing sequence tends to $+\infty$.
4. Let the function $f$ defined on $\mathbb{R}$ by $f(x) = \ln(x^{2} + 2x + 2)$. Statement: The function $f$ is convex on the interval $[-3; 1]$.
5. We consider the function mystery defined below which takes a list L of numbers as a parameter. We recall that len(L) returns the length, that is, the number of elements in the list $L$. \begin{verbatim} def mystery(L) : M = L[0] # We initialize M with the first element of the list L for i in range(len(L)) : if L[i] > M : M = L[i] return M \end{verbatim} Statement: The execution of mystery$([2, 3, 7, 0, 6, 3, 2, 0, 5])$ returns 7.

Exercise 3

Answer TRUE or FALSE to each of the following statements and justify your answer. Any answer without justification will not be taken into account in the grading. All questions in this exercise are independent.

1. Consider the sequence $( u _ { n } )$ defined for every non-zero natural number $n$ by $$u _ { n } = \frac { 25 + ( - 1 ) ^ { n } } { n }$$ Statement 1: The sequence $\left( u _ { n } \right)$ is divergent.
   
2. Consider the sequence $\left( w _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{aligned} w _ { 0 } & = 1 \\ w _ { n + 1 } & = \frac { w _ { n } } { 1 + w _ { n } } \end{aligned} \right.$
   It is admitted that for every natural number $n , w _ { n } > 0$. Consider the sequence $( t _ { n } )$ defined for every natural number $n$ by $t _ { n } = \frac { k } { w _ { n } }$ where $k$ is a strictly positive real number. Statement 2: The sequence $\left( t _ { n } \right)$ is a strictly increasing arithmetic sequence.
   
3. Consider the sequence $\left( v _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{array} { l l l } v _ { 0 } & = 1 \\ v _ { n + 1 } & = & \ln \left( 1 + v _ { n } \right) \end{array} \right.$ It is admitted that for every natural number $n , v _ { n } > 0$. Statement 3: The sequence $( v _ { n } )$ is decreasing.
   
4. Consider the sequence $\left( I _ { n } \right)$ defined for every natural number $n$ by $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } [ \ln ( x ) ] ^ { n } \mathrm {~d} x$.
   Statement 4: For every natural number $n , I _ { n + 1 } = \mathrm { e } - ( n + 1 ) I _ { n }$.

For each of the following statements, specify whether it is true or false then justify the answer given. Any answer without justification will not be taken into account.

1. Statement 1: Any decreasing sequence and bounded below by 0 converges to 0.
2. We consider a sequence $(u_n)$ defined on $\mathbb{N}$ such that, for every integer $n$, we have $$u_n \leq \frac{-9^n + 3^n}{7^n}.$$ Statement 2: $\lim_{n \rightarrow +\infty} u_n = -\infty$.
3. We consider the following function written in Python language: \begin{verbatim} def terme(N) : U = 1 for i in range(N) : U = U + i return U \end{verbatim} Statement 3: terme(4) returns the value 7.
4. During a competition, the winner has a choice between two prizes:
   • Prize A: they receive 1000 euros per day for 15 days;
   • Prize B: they receive 1 euro on day 1, 2 euros on day 2, 4 euros on day 3 and for 15 days the sum received doubles each day.
   Statement 4: The value of prize A is higher than the value of prize B.
5. We consider the sequence $(v_n)$ defined for every integer $n \geq 1$ by $$v_n = \int_1^n \ln x \mathrm{~d}x.$$ Statement 5: The sequence $(v_n)$ is increasing.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Let $( u _ { n } )$ be a sequence defined for all natural integer $n$ and satisfying the following relation: $$\text{for all natural integer } n , \frac { 1 } { 2 } < u _ { n } \leqslant \frac { 3 n ^ { 2 } + 4 n + 7 } { 6 n ^ { 2 } + 1 } .$$ Statement 1: $\lim _ { n \rightarrow + \infty } u _ { n } = \frac { 1 } { 2 }$.
   
2. Let $h$ be a function defined and differentiable on the interval $[-4;4]$. The graphical representation $\mathscr { C } _ { h ^ { \prime } }$ of its derivative function $h ^ { \prime }$ is given below. Statement 2: The function $h$ is convex on $[ - 1 ; 3]$.
   
3. The code of a building is composed of 4 digits (which may be identical) followed by two distinct letters among A, B and C (example: 1232BA). Statement 3: There exist 20634 codes that contain at least one 0.
   
4. We consider the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = x \ln x$. Statement 4: The function $f$ is a solution on $] 0 ; + \infty [$ of the differential equation $$x y ^ { \prime } - y = x .$$

Exercise 3 (5 points)

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions in this exercise are independent.

1. Consider a sequence ( $t _ { n }$ ) satisfying the recurrence relation: $$\text { for all natural integer } n , t _ { n + 1 } = - 0.8 t _ { n } + 18 .$$ Statement 1: The sequence ( $w _ { n }$ ) defined for all natural integer $n$ by $w _ { n } = t _ { n } - 10$ is geometric.
   
2. Consider a sequence ( $S _ { n }$ ) that satisfies for all non-zero natural integer $n$: $$3 n - 4 \leqslant S _ { n } \leqslant 3 n + 4 .$$ The sequence ( $u _ { n }$ ) is defined, for all non-zero natural integer $n$, by: $u _ { n } = \frac { S _ { n } } { n }$. Statement 2: The sequence ( $u _ { n }$ ) converges.
   
3. Consider the sequence $\left( v _ { n } \right)$ defined by: $$v _ { 1 } = 2 \text { and for all natural integer } n \geqslant 1 , v _ { n + 1 } = 2 - \frac { 1 } { v _ { n } } .$$ Statement 3: For all natural integer $n \geqslant 1 , v _ { n } = \frac { n + 1 } { n }$.
   
4. Consider the sequence ( $u _ { n }$ ) defined for all natural integer $n$ by $u _ { n } = \mathrm { e } ^ { n } - n$. Statement 4: The sequence $\left( u _ { n } \right)$ converges.
   
5. Consider the sequence ( $u _ { n }$ ) defined using the script written below in Python language, which returns the value of $u _ { n }$. \begin{verbatim} def u(n) : valeur = 2 for k in range(n) : valeur = 0.5 * (valeur + 2/valeur) return valeur \end{verbatim} We admit that ( $u _ { n }$ ) is decreasing and satisfies for all natural integer $n$: $$\sqrt { 2 } \leqslant u _ { n } \leqslant 2 .$$ Statement 5: The sequence $\left( u _ { n } \right)$ converges to $\sqrt { 2 }$.

3. Let $(v_n)$ be the sequence defined for every natural number $n$ by:
$$v_n = \frac{n}{2 + \cos(n)}.$$
Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
4. In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider points $A(1; 1; 2), B(5; -1; 8)$ and $C(2; 1; 3)$. Statement 4: $\overrightarrow{AB} \cdot \overrightarrow{AC} = 10$ and a measure of angle $\widehat{BAC}$ is $30°$.

Let $x _ { n }$ be a sequence with the following property: Every subsequence of $x _ { n }$ has a further subsequence which converges to $x$. Then the sequence $x _ { n }$ converges to $x$.

Let $f : ( 0 , \infty ) \longrightarrow \mathbb { R }$ be a continuous function. Then $f$ maps any Cauchy sequence to a Cauchy sequence.

Let $K \subset \mathbb { R } ^ { n }$ such that every real valued continuous function on $K$ is bounded. Then $K$ is compact (i.e closed and bounded).

If $A \subset \mathbb { R } ^ { 2 }$ is a countable set, then $\mathbb { R } ^ { 2 } \backslash A$ is connected.

The set $A = \left\{ ( z , w ) \in \mathbb { C } ^ { 2 } \mid z ^ { 2 } + w ^ { 2 } = 1 \right\}$ is bounded in $\mathbb { C } ^ { 2 }$.

Let $f , g : \mathbb { C } \longrightarrow \mathbb { C }$ be complex analytic, and let $h : [ 0,1 ] \longrightarrow \mathbb { C }$ be a non-constant continuous map. Suppose $f ( z ) = g ( z )$ for every $z \in \operatorname { Im } h$, then $f = g$. (Here $\operatorname { Im } h$ denotes the image of the function $h$.)

There is a non-constant continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ whose image is contained in $\mathbb { Q }$.

Suppose $f : \mathbb { R } \mapsto \mathbb { R } ^ { n }$ be a differentiable mapping satisfying $\| f ( t ) \| = 1$ for all $t \in \mathbb { R }$. Show that $\left\langle f ^ { \prime } ( t ) , f ( t ) \right\rangle = 0$ for all $t \in \mathbb { R }$. (Here $\|$.$\|$ denotes standard norm or length of a vector in $\mathbb { R } ^ { n }$, and $\langle . , .\rangle$ denotes the standard inner product (or scalar product) in $\mathbb { R } ^ { n }$.)

Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.

Let $f : X \mapsto Y$ be continuous map onto $Y$, and let $X$ be compact. Also $g : Y \mapsto Z$ is such that $g \circ f$ is continuous. Show $g$ is continuous.

Show that a biholomorphic map of the unit ball onto itself which fixes the origin is necessarily a rotation.

What can be said about 1-periodic sequences?

Exercise IV

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ be a sequence such that $u _ { n } \neq 0$ for every natural number $n$. For every natural number $n$, the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ is defined by $v _ { n } = - \frac { 2 } { u _ { n } }$. IV-A- If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is bounded below by $2$, then $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ is bounded below by $-1$. IV-B- If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is increasing, then $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ is decreasing. IV-C- If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ converges, then $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ converges.
For each statement, indicate whether it is TRUE or FALSE.

Verify that the converse of (Cesàro) is not always true by exhibiting a sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb{N} }$ that does not converge and such that $\left( \sigma _ { n } \right) _ { n \in \mathbb { N } }$ converges in $\mathbb { R }$.

2. For ALL APPLICANTS.

The functions $f ( n )$ and $g ( n )$ are defined for positive integers $n$ as follows:
$$f ( n ) = 2 n + 1 , \quad g ( n ) = 4 n$$
This question is about the set $S$ of positive integers that can be achieved by applying, in some order, a combination of $f _ { \mathrm { s } }$ and $g \mathrm {~s}$ to the number 1 . For example as
$$g f g ( 1 ) = g f ( 4 ) = g ( 9 ) = 36$$
and
$$f f g g ( 1 ) = f f g ( 4 ) = f f ( 16 ) = f ( 33 ) = 67$$
then both 36 and 67 are in $S$.
(i) Write out the binary expansion of 100 (one hundred). [0pt] [Recall that binary is base 2. Every positive integer $n$ can be uniquely written as a sum of powers of 2, where a given power of 2 can appear no more than once. So, for example, $13 = 2 ^ { 3 } + 2 ^ { 2 } + 2 ^ { 0 }$ and the binary expansion of 13 is 1101 .]
(ii) Show that 100 is in $S$ by describing explicitly a combination of $f _ { \mathrm { S } }$ and $g$ s that achieves 100 .
(iii) Show that 200 is not in $S$.
(iv) Show that, if $n$ is in $S$, then there is only one combination of applying $f _ { \mathrm { s } }$ and $g$ s in order to achieve $n$. (So, for example, 67 can only be achieved by applying $g$ then $g$ then $f$ then $f$ in that order.)
(v) Let $u _ { k }$ be the number of elements $n$ of $S$ that lie in the range $2 ^ { k } \leqslant n < 2 ^ { k + 1 }$. Show that
$$u _ { k + 2 } = u _ { k + 1 } + u _ { k }$$
for $k \geqslant 0$.
(vi) Let $s _ { k }$ be the number of elements $n$ of $S$ that lie in the range $1 \leqslant n < 2 ^ { k + 1 }$. Show that
$$s _ { k + 2 } = s _ { k + 1 } + s _ { k } + 1$$
for $k \geqslant 0$.
This page has been intentionally left blank

10. Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of real numbers satisfying $a_{n+1} = \frac{n(n+1)}{2} - a_{n}$ for all positive integers $n$. Which of the following options are correct?
(1) If $a_{1} = 1$, then $a_{2} = 1$
(2) If $a_{1}$ is an integer, then every term of the sequence is an integer
(3) If $a_{1}$ is irrational, then every term of the sequence is irrational
(4) $a_{2} \leq a_{4} \leq \cdots \leq a_{2n} \leq \cdots$ (where $n$ is a positive integer)
(5) If $a_{k}$ is odd, then $a_{k+2}, a_{k+4}, \ldots, a_{k+2n}, \ldots$ are all odd (where $n$ is a positive integer)

For a real number $a$, let $[a]$ denote the greatest integer not exceeding $a$. For example: $[1.2] = [\sqrt{2}] = 1$, $[-1.2] = -2$. Consider the irrational number $\theta = \sqrt{10001}$. Select the correct options.
(1) $a - 1 < [a] \leq a$ holds for all real numbers $a$
(2) The sequence $b_{n} = \frac{[n\theta]}{n}$ diverges, where $n$ is a positive integer
(3) The sequence $c_{n} = \frac{[-n\theta]}{n}$ diverges, where $n$ is a positive integer
(4) The sequence $d_{n} = n\left[\frac{\theta}{n}\right]$ diverges, where $n$ is a positive integer
(5) The sequence $e_{n} = n\left[\frac{-\theta}{n}\right]$ diverges, where $n$ is a positive integer

There is a real number sequence $\left\langle a_{n} \right\rangle$, where $a_{n} = \cos\left(n\pi - \frac{\pi}{6}\right)$, and $n$ is a positive integer. Select the correct options.
(1) $a_{1} = -\frac{1}{2}$
(2) $a_{2} = a_{3}$
(3) $a_{4} = a_{24}$
(4) $\left\langle a_{n} \right\rangle$ is a convergent sequence, and $\lim_{n \rightarrow \infty} a_{n} < 1$
(5) $\sum_{n=1}^{\infty} \left(a_{n}\right)^{n} = 3 - 2\sqrt{3}$