Let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.
We fix $\ell \in \mathbf { N }$ and $x \in [ 0,1 [$. Using the result of the previous question, establish the bound $\sum _ { n = 0 } ^ { \ell } p _ { n } x ^ { n } \leq P ( x )$. Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } z ^ { n }$.

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
We fix $\ell \in \mathbf{N}$ and $x \in [0,1[$. Using the result of the previous question, establish the bound $\sum_{n=0}^{\ell} p_n x^n \leq P(x)$. Deduce the radius of convergence of the power series $\sum_n p_n z^n$.

We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity, that is, $\lambda^n \neq 1$ for all integer $n \geqslant 1$. We propose to show that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$. Show that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.
Let $z \in D$. By examining the difference $\sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n } - \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
Let $z \in D$. By examining the difference $\sum_{n=0}^{+\infty} p_n z^n - \sum_{n=0}^{+\infty} p_{n,N} z^n$, prove that $$P(z) = \sum_{n=0}^{+\infty} p_n z^n$$

We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity. We have shown in question (19) that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$. Conclude that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$.

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. Show that there exists a sequence of polynomials $\left( Q _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ that converges uniformly towards $f$ on $I$ and such that, for all $n \in \mathbb { N } ^ { * }$, the function $Q _ { n }$ does not coincide with $f$ at any point of $I$, except possibly at zero: $$\forall n \in \mathbb { N } ^ { * } , \quad \forall x \in I \backslash \{ 0 \} , \quad Q _ { n } ( x ) \neq \exp ( x ).$$

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
Using the inequality from question 19, prove the inequality $$(4) \quad b(M) \leq e n b^{\prime}(M).$$

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Show that there exists $\omega > 0$ such that $|\lambda^m - \lambda| \geqslant \omega$ for all integer $m \geqslant 2$.

We consider a power series $f = \lambda I + F$ with $F \in O_2, \rho(F) > 0$. We still assume that $\lambda$ has modulus 1 and is not a root of unity. We consider the real $r_0 > 0$ given by question (24) (applied for $m = 1$) and the sequence $r_k$ defined by recursion from $r_0$ by the relation $$r_{k+1} = (1 - \alpha_{2^k})(1 + \alpha_{2^k}^2)^{-1}(1 + \alpha_{2^k})^{-1} \gamma_{2^k} r_k.$$ Show that there exist sequences $F_k$ and $P_k$ of elements of $O_2$, defined for $k \geqslant 0$, such that $F_0 = F$ and, for all $k \geqslant 0$, $$\begin{aligned} & \lambda I + F_{k+1} = (I + P_k)^\dagger \circ (\lambda I + F_k) \circ (I + P_k), \\ & F_k \in O_{1+2^k}, \quad P_k \in O_{1+2^k}, \\ & \widehat{F_k}(r_k) \leqslant r_k, \quad \widehat{P_k}(r_{k+1}) \leqslant r_k - r_{k+1}. \end{aligned}$$

We set $r_\infty := \lim r_k$ and $$h_k := (I + P_0) \circ (I + P_1) \circ \cdots \circ (I + P_{k-1}).$$ Explain why $r_\infty$ is well defined, and show that $\hat{h}_k(r_k) \leqslant r_0$ for all $k \geqslant 1$. Deduce that the series $h$ of question E satisfies $\hat{h}(r_\infty) \leqslant r_0$, thus that $\rho(h) \geqslant r_\infty$.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$.
15a. Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$.
15b. Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$.
Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by: $$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$ Deduce $f''(0)$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \in \mathbb{N}$ and $n \geqslant k+2$, we have
$$\left\|\psi_{n+1}^{(k)} - \psi_{n}^{(k)}\right\|_{\infty} \leqslant \frac{\mu_{n+1}}{2} \left\|\psi_{k+1}^{(k+1)}\right\|_{\infty}.$$
Deduce that for all $k \in \mathbb{N}$, the sequence of functions $\psi_{n}^{(k)}$ converges uniformly on $\mathbb{R}$.

Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that the limit $f = \lim_{n \rightarrow \infty} \psi_{n}$ is of class $C^{\infty}$, and that for all $k \geqslant 0$ we have
$$\left\|f^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{k}^{(k)}\right\|_{\infty}.$$

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) > 0$. Show that there exists $\rho _ { f } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { f } \leqslant \rho$ and such that $f > 0$ on $U _ { \rho _ { f } }$ and $\sqrt { f } \in \mathscr { D } _ { \rho _ { f } } ( \mathbb { R } )$.

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.

Mathematics Specialty - EXERCISE I (20 points)

First Part

Consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 2$ and for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined for every positive real $x$ by $f ( x ) = \frac { 3 x + 2 } { x + 4 }$. We admit that, for every natural number $\boldsymbol { n }$, $\boldsymbol { u } _ { \boldsymbol { n } }$ is greater than or equal to $1$.
I-1-a- Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$. Give the result as an irreducible fraction.
I-1-b- The graph below gives the representative curve in an orthonormal coordinate system of the function $f$. From this graph, what can be conjectured about the variations and convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$? Specify the possible limit.

Second Part - Method 1

I-2-a- Show that, for every natural number $n$, $u _ { n + 1 } - u _ { n } = \frac { \left( 1 - u _ { n } \right) \left( u _ { n } + 2 \right) } { u _ { n } + 4 }$. I-2-b- Deduce the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify your answer. I-3- Prove that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. Let $l$ denote its limit. I-4- Determine the value of $l$. Justify your answer.

Third Part - Method 2

Consider the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ defined for every natural number $n$ by: $v _ { n } = \frac { u _ { n } - 1 } { u _ { n } + 2 }$. I-5- Calculate $v _ { 0 }$. I-6-a- Determine the constant $k$ in $] 0 ; 1 [$ such that $v _ { n + 1 } = k \times v _ { n }$ for every natural number $n$. Justify your answer. What can be deduced about the nature of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$? For questions $\mathbf { I - 6 - b }$ and $\mathbf { I - 6 - c }$, answers may be expressed as a function of $k$ or its value. I-6-b- Deduce the expression of $v _ { n }$ as a function of $n$. I-6-c- Deduce the convergence of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer. I-7-a- Express $u _ { n }$ as a function of $v _ { n }$ for every natural number $n$. I-7-b- Deduce the convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \right)$$ where $\sigma_n = \frac{1}{n+1}\sum_{k=0}^n u_k$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ takes real values, prove that the result holds if $\ell = + \infty$ or $\ell = - \infty$.

Using (Cesàro), calculate the limit of the sequence $\left( v _ { n } \right) _ { n \geqslant 1 }$ defined by $v _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k n }$. Then, using a series-integral comparison, give an equivalent of $\left( v _ { n } \right) _ { n \geqslant 1 }$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ and $\alpha \in \mathbb { R } ^ { * }$. Suppose that $\lim _ { n \rightarrow + \infty } e _ { n } = \alpha$, where $e_n = u_{n+1} - u_n$. Using (Cesàro), give an equivalent of $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Recover this result using a comparison theorem for series with positive terms.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \left] 0 , + \infty \right[ ^ { \mathbb { N } }$ and $\ell \in \left] 0 , + \infty \right[$. Suppose that $\lim _ { n \rightarrow + \infty } \frac { u _ { n + 1 } } { u _ { n } } = \ell$. Prove that $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { u _ { n } } = \ell$. Prove that the result holds if $\ell = 0$ or $\ell = + \infty$. Deduce $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { n ! }$ and $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { \frac { n ^ { n } } { n ! } }$.