Show that $S$ is continuous on $\mathbf{R}_+$.
Hint: You may first show that, if $x \in \mathbf{R}$, $t \mapsto P_t(f)(x)$ is continuous on $\mathbf{R}_+$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that a symmetric matrix $D$ of order $n$ with non-negative coefficients and zero diagonal is EDM if and only if $-\frac{1}{2}PDP$ is positive.

We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$.
Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$

Give a new proof, based on questions 12 and 13 above, of the fact that the power series $\sum_{m=0}^{\infty} x^{m^2}$ is not the expansion of a rational function.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$.
Let $s > 0$. We define the functions $f$ and $g _ { s }$ on $\overline { U _ { n } }$ by setting, for all $x = \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } }$,
$$f ( x ) = \prod _ { k = 1 } ^ { n } x _ { k } \quad \text { and } \quad g _ { s } ( x ) = \left( \sum _ { k = 1 } ^ { n } x _ { k } \right) - s .$$
We denote by $X _ { s }$ the subset of $\overline { U _ { n } }$ consisting of the zeros of $g _ { s } : X _ { s } = \left\{ x \in \overline { U _ { n } } \mid g _ { s } ( x ) = 0 \right\}$.
Prove that, for all $\left( x _ { 1 } , \ldots , x _ { n } \right) \in U _ { n } \cap X _ { s } , \left( \prod _ { i = 1 } ^ { n } x _ { i } \right) ^ { 1 / n } \leqslant \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i }$ and deduce the arithmetic-geometric inequality
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbb { R } _ { + } \right) ^ { n } , \quad \left( \prod _ { i = 1 } ^ { n } x _ { i } \right) ^ { 1 / n } \leqslant \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i }$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and $J(\tau, R) = \sum_{i=1}^{n} |\boldsymbol{y}_{i} - (R\boldsymbol{x}_{i} + \tau)|^{2}$. For all $R \in \mathrm{SO}_{d}(\mathbb{R})$, $\tau(R)$ denotes the unique minimizer of $\tau \mapsto J(\tau, R)$.

• [(a)] Show that there exists $R_{*} \in \mathrm{SO}_{d}(\mathbb{R})$ such that $J(\tau(R_{*}), R_{*}) \leqslant J(\tau, R)$ for all $(\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$.
• [(b)] Show that $R_{*}$ is not necessarily unique.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$, $B_{n}' = n B_{n-1}$.

Verify that we have $S(0) = \operatorname{Ent}_{\varphi}(f)$ and $\lim_{t \rightarrow +\infty} S(t) = 0$, where $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$.

Deduce that:
$$\int _ { \frac { \pi } { 2 } } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \left( \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } t \sin ( t ) } { t ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } \right) \mathrm { d } t$$

Verify that we have $S(0) = \operatorname{Ent}_{\varphi}(f)$ and $\lim_{t \rightarrow +\infty} S(t) = 0$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that every non-zero symmetric matrix with non-negative coefficients and zero diagonal, having a unique strictly positive eigenvalue with eigenspace of dimension 1 and eigenvector $\mathbf{e}$, is EDM.

Let $\left(a_{n}\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_{k}$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^{1}$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_{k} b(k) = A(n) b(n) - \int_{2}^{n} b^{\prime}(t) A(t) \mathrm{d}t.$$

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Let $\left(a_n\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_k$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^1$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_k b(k) = A(n)b(n) - \int_2^n b'(t) A(t) \, \mathrm{d}t$$

Using the Bernoulli polynomials $(B_n)_{n \in \mathbb{N}}$ satisfying $B_n(z+1) - B_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$, deduce the expression of a polynomial function satisfying the equation $(E_h)$: $$\forall x \in \mathbb{C},\, f(x+1) - f(x) = h(x)$$ on $\mathbb{C}$ when $h$ is a polynomial function.

Let $(c_n)_{n \in \mathbf{N}^*}$ and $(d_n)_{n \in \mathbf{N}^*}$ be two sequences of strictly positive real numbers such that: $c_n \underset{n \to +\infty}{\sim} d_n$ and the series $\sum_n c_n$ diverges.
We admit without proof the following result:
Theorem 1. Let $(a_n)_{n \in \mathbf{N}^*}$ and $(b_n)_{n \in \mathbf{N}^*}$ be two sequences of nonzero real numbers such that $a_n = o(b_n)$ as $n \to +\infty$ and the series $\sum_n |b_n|$ is divergent. Then: $$\sum_{k=1}^n a_k = o\!\left(\sum_{k=1}^n |b_k|\right) \text{ as } n \to +\infty.$$
By using this result, show that the series $\sum_n d_n$ is divergent and that: $$\sum_{k=1}^n c_k \underset{n \rightarrow +\infty}{\sim} \sum_{k=1}^n d_k.$$

Show that $(B_{n})_{n \in \mathbb{N}}$ is the unique sequence of polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$

Let $r \geq 2$ be an integer and $a_1, \ldots, a_r \in \mathbf{Q}$ be distinct rationals. Let $b_1, \ldots, b_r \in \mathbf{Q}^{\times}$ be nonzero rationals. Set $e^{a_i x} \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{a_i^n}{n!} x^n$ and consider the power series $$f(x) = \sum_{n=0}^{\infty} \frac{u_n}{n!} x^n \stackrel{\text{def}}{=} b_1 e^{a_1 x} + \cdots + b_r e^{a_r x}.$$ Show that the Laplace transform $\widehat{f}(x) = \sum_{n=0}^{\infty} u_n x^n$ is the power series expansion of the rational function $$\sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$ Deduce that $f$ is not the zero power series.

Let $(H_{n})_{n \in \mathbb{N}}$ be the sequence of polynomials defined by: $\forall n \in \mathbb{N},\, H_{n}(X) = (-1)^{n} B_{n}(1-X)$, where $(B_n)_{n\in\mathbb{N}}$ are the Bernoulli polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$ Show that for all $n \in \mathbb{N}$, $H_{n} = B_{n}$.

We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. The random variable $N_n$ counts the number of equality indices of the path $(X_1(\omega), \cdots, X_{2n}(\omega))$, and it has been shown that: $$\mathbb{E}(N_n) = \sum_{i=1}^n \frac{\binom{2i}{i}}{4^i}.$$ Deduce the equivalent: $$\mathbb{E}(N_n) \underset{n \to +\infty}{\sim} \frac{2}{\sqrt{\pi}} \sqrt{n}.$$

Consider the sequence $(v_n)_{n \geq 0}$ defined in terms of the coefficients $u_n$ by the formula $$v_n = n! \sum_{i=0}^{n} \frac{u_i}{i!}$$ and the power series $$v(x) = \sum_{n=0}^{\infty} v_n x^n \in \mathbf{Q}\llbracket x \rrbracket.$$ Show the equality of power series $$\sum_{n=0}^{\infty} (v_n - n v_{n-1}) x^n = \sum_{n=0}^{\infty} u_n x^n.$$

By comparison with an integral, establish that $$\sum_{k=1}^{n} \ln(k) \underset{n \rightarrow +\infty}{=} n\ln(n) - n + O(\ln(n))$$

Justify that the series $\sum_{k \geqslant 2} \frac{\ln(k)}{k(k-1)}$ converges.

By comparison with an integral, establish that $$\sum_{k=1}^{n} \ln(k) \underset{n \rightarrow +\infty}{=} n\ln(n) - n + O(\ln(n))$$