Give the value of $\nu_{\sigma}$.

We assume that $f \in \mathcal { C } ( [ 0,1 ] , \mathbb { R } )$ satisfies: $$\exists \alpha \in ]0,1] , \exists K \geq 0 , \forall ( y , z ) \in [ 0,1 ] ^ { 2 } , | f ( y ) - f ( z ) | \leq K | y - z | ^ { \alpha }$$ For all $n \in \mathbb{N}^*$, define $B _ { n } f ( X ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } X ^ { k } ( 1 - X ) ^ { n - k }$.
Let $n \in \mathbb { N } ^ { * }$. Show that
$$\left\| f - B _ { n } f \right\| _ { \infty } \leq \frac { 3 K } { 2 } \frac { 1 } { n ^ { \alpha / 2 } }$$
Hint: One may first express $f ( x ) - B _ { n } f ( x )$ in terms of $\mathbb { E } ( f ( x ) - f \left( S _ { n } \right) )$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Show that, for all $z \in \mathcal { D }$, the power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ converges.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { 0 } ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } z ^ { n }$ and, subject to convergence, $$\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$$
Justify that, for all $p \in \mathbb { N } ^ { * }$ and all $\left. x \in \right] - 1,1 \left[ , \Phi _ { p } ( x ) = \varphi ^ { ( p ) } ( x ) \right.$ and that for all $p \in \mathbb { N } ^ { * }$ and all $z \in \mathcal { D }$, the power series $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$ converges.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Justify that, for all $p \in \mathbb { N }$, the function $\varphi ^ { ( p ) }$ is bounded on $] - 1,1 [$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Let $r$ be a real number in the interval $] 0,1 [$. Demonstrate for all integers $n \geqslant 1$ and $p \geqslant 1$, that $$( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } r ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \Phi _ { p } \left( r \mathrm { e } ^ { \mathrm { i } \theta } \right) \mathrm { e } ^ { - n \mathrm { i } \theta } \mathrm {~d} \theta$$

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that $$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } a _ { n }$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$.
Let $p \in \mathbb { N } ^ { * }$. Show that the power series $\sum _ { n \geqslant 0 } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p }$.

The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that all moments of order $p$ of the sequence $(a _ { n })$ are zero.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Show that $\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } | \theta ( x ) | = 0$.

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that $$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that $\forall r \in [0, R[$, $|f(0)| \leqslant \sup_{t \in \mathbb{R}} |f(r\cos(t), r\sin(t))|$.

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that if $|f|$ attains a maximum at 0, then $f$ is constant on $D(0,R)$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z) \, \mathrm{d}t = 1$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $\delta \in ]0, \pi[$ and all real $\varphi$, $$\int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t \xrightarrow[z \rightarrow \mathrm{e}^{\mathrm{i}\varphi}]{} 0$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Using Heine's theorem, show that, for all $\varepsilon > 0$, there exists $\delta > 0$ such that, for all real number $\varphi$ and all complex number $z$ satisfying $|z| < 1$, $$|g(z) - h(\varphi)| \leqslant \frac{\sup_{t \in \mathbb{R}} |h(t)|}{\pi} \int_{\varphi+\delta}^{\varphi+2\pi-\delta} \mathcal{P}(t,z) \, \mathrm{d}t + \varepsilon$$

Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
State Cauchy's theorem for a scalar first-order linear differential equation and prove that, for all $x \in ]-1,1[$, $$f_{\alpha}(x) = \sum_{n=0}^{+\infty} L_{n}(\alpha) \frac{x^{n}}{n!}.$$

Let $\alpha_n = f^{(n)}(0)$ where $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$. Let $R$ be the radius of convergence of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ and $g$ its sum. Using Taylor's formula with integral remainder, show $$\forall N \in \mathbb{N}, \forall x \in \left[0, \pi/2\left[, \quad \sum_{n=0}^{N} \frac{\alpha_n}{n!} x^n \leqslant f(x)\right.\right.$$

Using the result of Q6, deduce the lower bound $R \geqslant \pi/2$ for the radius of convergence $R$ of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$.

Using the fact that $f(x) = g(x)$ on $I = ]-\pi/2, \pi/2[$ where $f(x) = \frac{\sin x + 1}{\cos x}$ and $g$ is the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$, deduce that $R = \pi/2$.

Using the decomposition of $f(x) = g(x) = \frac{\sin x + 1}{\cos x}$ into even and odd parts, deduce $$\forall x \in I, \quad \tan(x) = \sum_{n=0}^{+\infty} \frac{\alpha_{2n+1}}{(2n+1)!} x^{2n+1} \quad \text{and} \quad \frac{1}{\cos x} = \sum_{n=0}^{+\infty} \frac{\alpha_{2n}}{(2n)!} x^{2n}.$$

Let $t$ be the function defined on $I = ]-\pi/2, \pi/2[$ by $t(x) = \tan(x)$. For every natural integer $n$, express $t^{(n)}(0)$ as a function of the reals $(\alpha_i)_{i \in \mathbb{N}}$.