Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$. We set: $d _ { k } ( \epsilon ) = \sup _ { t \in [ \epsilon , 2 \pi - \epsilon ] } R _ { k } ( t )$. Prove then that $$\lim _ { k \rightarrow + \infty } d _ { k } ( \epsilon ) = 0$$

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde{a}_n = \frac{A_n}{n+1}$ where $A_n = \sum_{k=0}^n a_k$.
Conclude (i.e., prove property II.3: $\lim_{n \to \infty} \widetilde{a}_n = 1$).

We use the notation $R$ introduced in part I. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$. We denote $$r = \left|R\left(z^2 - 1\right)\right|, \quad s = \left|z + R\left(z^2 - 1\right)\right|, \quad t = \left|z - R\left(z^2 - 1\right)\right|, \quad h = \max(s,t)$$ We also denote $V_n(z) = U_{n+1}(z,-1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$. Prove that, for all $n \in \mathbb{N}$, $$\left|V_n(z)\right| \leqslant \frac{h^{n+1}}{r}$$

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 set $m=f'(1)$. We consider the recurrent sequence $(u_n)_{n\in\mathbb{N}}$ defined by $u_0=0$ and, for all $n\in\mathbb{N}$, $u_{n+1}=f(u_n)$.
We now assume $m<1$ and we set again, for $n\in\mathbb{N}$, $\varepsilon_n=1-u_n$.
Deduce that there exists $c>0$ such that, as $n$ tends to infinity, $1-u_n\sim cm^n$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q12, deduce an asymptotic equivalent of $f$ at $+\infty$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Using the fact that $0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}$ for all $t \in \mathbb{R}$, deduce that $$\forall x \in [0, \mathrm{e}], \quad \forall n \in \mathbb{N}, \quad |w_n(x) - W(x)| \leqslant \left(\frac{x}{\mathrm{e}}\right)^n |1 - W(x)|.$$

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 $m \geqslant 2$.

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 $$(4) \quad b(M) \leq e n b^{\prime}(M).$$

Show that there exists a real number $c_1 > 0$ such that: for all $n \geq 0$, $$|v_n| \leq c_1 \frac{A^{n+1}}{n+1}.$$

Problem 2, Part 3: Linear recurrence sequences with variable coefficients
We consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form $$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$ where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \} , \left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0 , V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied (all complex roots of $P(X) = X^d - \sum_{i=0}^{d-1} a_i X^i$ have modulus strictly less than 1), and $A$ is the matrix from question 7.
Let $\varepsilon > 0$ be fixed. Show that there exists an integer $q \geqslant 1$ and an integer $n _ { 0 }$ such that for all $n \geqslant n _ { 0 }$, $$\left\| V _ { n + q } \right\| _ { \infty } \leqslant ( \sigma ( A ) + \varepsilon ) ^ { q } \left\| V _ { n } \right\| _ { \infty }$$ where $A$ is the matrix from question 7.