Let $A > 2$. Show that, for every $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

Let $A > 2$. Show that, for all $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

Using the results of questions 28--30, show that there exists a unique complex sequence $(b_n)_{n \in \mathbb{N}}$ and a real number $r > 0$ such that, for all $z \in \mathbb{C}$, $$0 < |z| < r \Rightarrow \frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n.$$

Let $A > 2$. Show that $$\lim_{n \rightarrow +\infty} \frac{1}{n} \mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) = 0.$$

Let $A > 2$. Show that $$\lim_{n \rightarrow +\infty} \frac{1}{n} \mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) = 0.$$

Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$, with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ is an orthogonal system and that, for all $n \in \mathbb { N } , \left\| U _ { n } \right\| = 1$. To calculate the value of $( U _ { m } \mid U _ { n } )$, one may perform the change of variable $x = \cos ^ { 2 } \theta$.

Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ (valid for $0 < |z| < r$), by performing a Cauchy product, show that $b_0 = 1$ and, for all integer $n \geqslant 2$, $$\sum_{p=0}^{n-1} \binom{n}{p} b_p = 0.$$

Let $A > 2$. Let $f$ be a continuous and bounded function from $\mathbb{R}$ to $\mathbb{R}$ and $P$ a polynomial of degree $p$. Justify that there exists a constant $K$ such that $$\forall x \in \mathbb{R} \setminus ]-A, A[, \quad |f(x) - P(x)| \leqslant K|x|^{p}.$$

Let $A > 2$. Let $f$ be a continuous and bounded function from $\mathbb{R}$ to $\mathbb{R}$ and $P$ a polynomial of degree $p$. Justify that there exists a constant $K$ such that $$\forall x \in \mathbb{R} \setminus \left]-A, A\right[, \quad |f(x) - P(x)| \leqslant K|x|^{p}.$$

Using the relation $b_0 = 1$ and $\sum_{p=0}^{n-1} \binom{n}{p} b_p = 0$ for all integer $n \geqslant 2$, deduce the value of $b_1, b_2, b_3$ and $b_4$.

Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$.
Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^\star$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$

Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^*$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$

For all $n \in \mathbb { N }$, set $\mu _ { n } = \left( X ^ { n } \mid 1 \right)$ with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\forall n \in \mathbb { N } , \mu _ { n } = C _ { n }$.

Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ and a parity argument, show that $b_{2p+1} = 0$ for all integer $p \geqslant 1$.

Let $n \in \mathbb { N }$. Deduce from the previous parts the value of the determinant $$H _ { n } = \operatorname { det } \left( C _ { i + j - 2 } \right) _ { 1 \leqslant i , j \leqslant n + 1 } = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \ddots & \therefore & \vdots \\ \vdots & \therefore & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & \therefore & \ddots & & \therefore & C _ { 2 n - 1 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 2 } & C _ { 2 n - 1 } & C _ { 2 n } \end{array} \right|.$$

The polynomials $B_m$ are defined by $$\forall m \in \mathbb{N}, \quad B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}.$$
Determine $B_0, B_1, B_2$ and $B_3$.

We set, for all $n \in \mathbb { N }$, $$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$ with the inner product $( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x$. Let $( n , k ) \in \mathbb { N } ^ { 2 }$ such that $k < n$. Show $\left( D _ { n } \mid X ^ { k } \right) = 0$.

The polynomials $B_m$ are defined by $$\forall m \in \mathbb{N}, \quad B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}.$$
Show that, for all integer $m \geqslant 2$, $B_m(1) = b_m$, then that, for all integer $m \geqslant 1$, $B_m' = m B_{m-1}$.

We set, for all $n \in \mathbb { N }$, $$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$ and $$H _ { n } ^ { \prime } = \left| \begin{array} { c c c c c c } C _ { 1 } & C _ { 2 } & C _ { 3 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 2 } & \therefore & & \ddots & \ddots & C _ { n + 1 } \\ C _ { 3 } & & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 3 } \\ C _ { n - 1 } & \therefore & \ddots & & \ddots & C _ { 2 n - 2 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 3 } & C _ { 2 n - 2 } & C _ { 2 n - 1 } \end{array} \right|.$$ Deduce that $\forall n \in \mathbb { N } , D _ { n } = U _ { n }$, then determine, for all $n \in \mathbb { N } ^ { * }$, the value of the determinant $H _ { n } ^ { \prime }$.

We fix an integer $n \in \mathbb{N}^*$ and consider a function $g : [0,n] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$. The polynomial $B_1$ is as defined in the sequence $(B_m)$.
Show that $$\int_0^n g(x)\,\mathrm{d}x = \sum_{k=0}^{n-1} \frac{g(k) + g(k+1)}{2} - \int_0^n B_1(x - \lfloor x \rfloor) g'(x)\,\mathrm{d}x.$$

We fix an integer $n \in \mathbb{N}^*$ and consider a function $g : [0,n] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$. The polynomials $B_m$ are defined by $B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}$.
Deduce that for all integer $m \geqslant 2$, $$\int_0^n g(x)\,\mathrm{d}x = \sum_{k=0}^{n-1} \frac{g(k)+g(k+1)}{2} + \sum_{p=2}^m \frac{(-1)^{p-1} b_p}{p!}\left(g^{(p-1)}(n) - g^{(p-1)}(0)\right) + \frac{(-1)^m}{m!} \int_0^n B_m(x - \lfloor x \rfloor) g^{(m)}(x)\,\mathrm{d}x.$$

Let $\ell$ be a strictly positive integer. We are given a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ of vectors in $\mathbb { R } ^ { \ell }$ such that the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$ converges.
(a) Show that the sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ is convergent.
(b) Let $v ^ { * }$ denote the limit of this sequence. Bound $\left\| v _ { n } - v ^ { * } \right\|$ by means of a remainder of the sum of the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent.

For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent, where $$g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right)$$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x).$$