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

With the notation of question 18 and 19, show that the differential operator $L = -x^2 \left(\frac{d}{dx}\right) + (1-x)$ acts on $v(x)$ by $$(L \cdot v)(x) = \sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Prove that $\omega _ { 1 } \leqslant \frac { 1 } { \mathrm { e } }$ and that, for all $k$ in $\llbracket 1 , n \rrbracket , \omega _ { k } \leqslant \frac { k } { k + 1 }$.
You may prove, for $k \in \llbracket 1 , n - 1 \rrbracket$, that $\omega _ { k + 1 } ^ { k + 1 } = \frac { 1 } { \lambda } \omega _ { k } ^ { k } \left( 1 - \frac { \omega _ { k } } { k } \right) ^ { - k }$.

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^{2}}$ is integrable on $[2, +\infty[$.

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Establish that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} \underset{n \rightarrow +\infty}{=} \ln_{2}(n) + c_{1} + O\left(\frac{1}{\ln(n)}\right)$, for a real $c_{1} \in \mathbb{R}$ to be determined.

We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^2}$ is integrable on $[2, +\infty[$.

We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Establish that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} \underset{n \rightarrow +\infty}{=} \ln_2(n) + c_1 + O\left(\frac{1}{\ln(n)}\right)$, for a real $c_1 \in \mathbb{R}$ to be determined.

In an urn containing $n$ white balls and $n$ black balls, we proceed to draw balls without replacement, until the urn is completely empty. The random variable $M_n$ counts the number of equality indices $k$ between $1$ and $2n$, and it has been shown that: $$\mathbb{E}(M_n) = \sum_{i=0}^{n-1} \frac{\binom{2i}{i} \cdot \binom{2n-2i}{n-i}}{\binom{2n}{n}}.$$ Deduce the equivalent: $$\mathbb{E}(M_n) \underset{n \to +\infty}{\sim} \sqrt{\pi n}.$$

With the notation of questions 18--20, deduce that if $v(x)$ is the power series expansion of a rational fraction $P/Q$, then every element of the non-empty set $\{1/a_i \mid a_i \neq 0\}$ is a pole of $P/Q$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Reach a contradiction on $\omega _ { n }$. Deduce that, for all $n$ in $\mathbb { N } ^ { * }$, for all $\left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$ such that $x _ { 1 } + \cdots + x _ { n } = 1$,
$$\sum _ { k = 1 } ^ { n } \left( x _ { 1 } x _ { 2 } \cdots x _ { k } \right) ^ { 1 / k } \leqslant \mathrm { e }$$

Deduce Carleman's inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } a _ { n }$$
for any sequence $\left( a _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ of strictly positive real numbers such that $\sum a _ { n }$ converges.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ and $\left( c _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be two sequences of strictly positive real numbers. Prove that
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \sum _ { k = 1 } ^ { + \infty } c _ { k } a _ { k } \sum _ { n = k } ^ { + \infty } \frac { 1 } { n } \left( \prod _ { i = 1 } ^ { n } c _ { i } \right) ^ { - 1 / n }$$

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ and $\left( c _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be two sequences of strictly positive real numbers. We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
By considering $c _ { n } = \frac { ( n + 1 ) ^ { n } } { n ^ { n - 1 } }$, deduce the Carleman-Yang inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } \left( 1 - \sum _ { k = 1 } ^ { + \infty } \frac { b _ { k } } { ( n + 1 ) ^ { k } } \right) a _ { n }$$

We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
The Carleman-Yang inequality states:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } \left( 1 - \sum _ { k = 1 } ^ { + \infty } \frac { b _ { k } } { ( n + 1 ) ^ { k } } \right) a _ { n }$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , b _ { n } \geqslant 0$. In what way is the previous inequality a refinement of Carleman's inequality?

Let $\ell \geq 0$ be an integer. Show that there exists a polynomial $P_\ell \in \mathbf{Q}[x]$ of degree $< r(\ell+1)$ satisfying $$\sum_{n=0}^{\infty} n(n-1)\cdots(n-\ell+1)\, u_{n-\ell}\, x^n = \frac{P_\ell(x)}{\left(1 - s_1 x - \cdots - s_r x^r\right)^{\ell+1}}.$$

Define two sequences $(w_{n,k})_{n,k \geq 0}$ and $(w_n(k))_{n,k \geq 0}$ by the formulas $$w_{n,k} = n! \sum_{i=0}^{n-k} \frac{u_i}{i!} \quad \text{and} \quad \sum_{n=0}^{\infty} w_n(k) x^n = \left(1 - s_1 x - \cdots - s_r x^r\right)^k \sum_{n=0}^{\infty} w_{n,k}\, x^n.$$ Show the equality $w_n(k) = v_n(k)$ for all $n$ and $k$ such that $n \geq kr$.

Show the equality $$v_n(k) = \sum_{i=1}^{r} b_i e^{a_i} \int_{a_i}^{\infty} e^{-t} t^{n-kr} (t - a_1)^k \cdots (t - a_r)^k\, dt.$$

Show that if $v_n(k)$ is non-zero and $n \geq kr$, then $$k! \leq |D^n v_n(k)| \leq c_2 (AD)^n C^k.$$

Deduce that there exists an integer $k_0$ such that $$v_n(k) = 0 \text{ for all } k \geq k_0 \text{ and } kr \leq n \leq 10kr.$$

Conclude that $v_n(k_0) = 0$ for all $n \geq k_0 r$.

Justify that, for all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, the series $\sum u _ { k }$ converges, where $u_k = \dfrac{(-1)^k}{pk+q}$.

Show that the set of quasi-polynomial functions forms a $\mathbb{C}$-vector space.

In this question, we set $p = q = 1$. Show that $$\phi _ { 1,1 } ( n ) = \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + t } d t - \int _ { 0 } ^ { 1 } \frac { ( - t ) ^ { n + 1 } } { 1 + t } d t$$ where $\phi_{1,1}(n) = \sum_{k=0}^{n} \dfrac{(-1)^k}{k+1}$.

Show that if $P, Q : \mathbb{Z} \rightarrow \mathbb{C}$ are two quasi-polynomial functions such that $P(n) = Q(n)$ for all $n \geq 0$, then $P = Q$.

Deduce the value of $S _ { 1,1 }$, the sum of the congruent-harmonic series with parameters $p=q=1$.