We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
III.B.1) For all $k \in \mathbb { N }$, establish a simplified expression for $H _ { k + 1 } ( X ) + k H _ { k } ( X )$.
III.B.2) Deduce that, for every natural integer $n$ $$X ^ { n } = \sum _ { k = 0 } ^ { n } S ( n , k ) H _ { k } ( X )$$

We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Let $k \in \mathbb { N }$.
III.C.1) Show that the function $f _ { k } : x \mapsto \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$ is defined on $] - 1,1 [$.
III.C.2) For $k \in \mathbb { N }$, we consider the function $g _ { k } : x \mapsto \frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! }$.
Show that the function $g _ { k }$ satisfies the differential equation $$y ^ { \prime } = \frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k - 1 } } { ( k - 1 ) ! } + k y$$
III.C.3) Deduce that for all $k \in \mathbb { N }$ and for all $x \in ] - 1,1 [$, $$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$

We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ It has been established that for all $k \in \mathbb { N }$ and $x \in ] - 1,1 [$, $$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$
III.D.1) For $x \in ] - 1,1 [$ and $\alpha \in \mathbb { R }$, simplify $\sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { x ^ { k } } { k ! }$.
III.D.2) Show that for $u < \ln 2$ $$\mathrm { e } ^ { u \alpha } = \sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { \left( \mathrm { e } ^ { u } - 1 \right) ^ { k } } { k ! }$$

We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$
Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.

We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ We define $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$, and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Deduce that $U _ { n } ( p ) = \sum _ { k = 0 } ^ { n } \frac { S ( n , k ) } { k + 1 } H _ { k + 1 } ( p + 1 )$.

We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ We denote $F = \left\{ P \in \mathbb { R } _ { n } [ X ] \mid P ( 0 ) = 0 \right\}$, then $G = \operatorname { Vect } \left( X ^ { 2 k + 1 } ; 0 \leqslant k \leqslant n - 1 \right)$.
Let $Q ( X )$ be the polynomial such that $\forall p \in \mathbb { N } , Q ( p ) = \sum _ { k = 0 } ^ { p } k$.
V.D.1) Recall the explicit expression of the polynomial $Q ( X )$.
V.D.2) Show that the map: $$\begin{aligned} \Phi : F & \rightarrow G \\ P ( X ) & \mapsto \Delta ( P ( Q ( X - 1 ) ) ) \end{aligned}$$ is an isomorphism.
V.D.3) Deduce that for all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that $$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$

We fix $n \in \mathbb { N }$. For all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that $$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$
V.E.1) Determine the leading term in $P _ { r } ( X )$.
V.E.2) Show that for $r \geqslant 1$, $X ^ { 2 }$ divides $P _ { r } ( X )$.
V.E.3) Explicitly give the polynomials $P _ { 1 } ( X )$ and $P _ { 2 } ( X )$.

Show that the set $E^{c}$ is non-empty.

Is the set $E^{c}$ a vector subspace of $\mathbb{R}^{\mathbb{N}}$?

Show that $E^{c}$ is strictly included in $E$.

Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be an element of $E^{c}$. Show that $\ell^{c}$ belongs to the segment $[0,1]$.

Let $k$ be a strictly positive integer and $q$ a real belonging to the interval $]0,1[$. Show that the sequences $\left(\frac{1}{(n+1)^{k}}\right)_{n \in \mathbb{N}},\left(n^{k} q^{n}\right)_{n \in \mathbb{N}}$ and $\left(\frac{1}{n !}\right)_{n \in \mathbb{N}}$ belong to $E^{c}$ and give their convergence rate.

We consider the sequence $\left(v_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, v_{n}=\left(1+\frac{1}{2^{n}}\right)^{2^{n}}$.
a) Show that in the neighbourhood of $+\infty, v_{n}=\mathrm{e}-\frac{\mathrm{e}}{2^{n+1}}+o\left(\frac{1}{2^{n}}\right)$.
b) Show that the sequence $(v_{n})$ belongs to $E^{c}$ and give its convergence rate.

We consider the sequence $\left(I_{n}\right)_{n \in \mathbb{N}}$ defined by $I_{0}=0$ and $\forall n \in \mathbb{N}^{*}, I_{n}=\int_{0}^{+\infty} \ln\left(1+\frac{x}{n}\right) \mathrm{e}^{-x} \mathrm{~d} x$.
a) Show that the sequence $(I_{n})$ is well defined and belongs to $E$.
b) Using integration by parts, show that the sequence $(I_{n})$ belongs to $E^{c}$ and give its convergence rate.

Let $\alpha$ be a real strictly greater than 1. The Riemann series $\sum_{n \geqslant 1} \frac{1}{n^{\alpha}}$ converges to a real that we will denote $\ell$. We denote by $\left(S_{n}\right)_{n \in \mathbb{N}}$ the sequence defined by $S_{0}=0$ and $\forall n \geqslant 1, S_{n}=\sum_{k=1}^{n} \frac{1}{k^{\alpha}}$.
a) Show that $\forall n \geqslant 1, \frac{1}{\alpha-1} \frac{1}{(n+1)^{\alpha-1}} \leqslant \ell-S_{n} \leqslant \frac{1}{\alpha-1} \frac{1}{n^{\alpha-1}}$.
b) Deduce that $\left(S_{n}\right)_{n \in \mathbb{N}}$ belongs to $E^{c}$ and give its convergence rate.

Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be an element of $E$ whose convergence rate is of order $r$, where $r$ is a real strictly greater than 1. Show that the convergence of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is fast.

a) Show that the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, S_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ is an element of $E$. We denote by $s$ the limit of this sequence.
b) Show that for every natural integer $n$, we have $\frac{1}{(n+1)!} \leqslant s-S_{n} \leqslant \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{2^{k}}$.
c) Deduce that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ is fast.
d) Let $r$ be a real strictly greater than 1. Show that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ towards $s$ is not of order $r$.

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ We define for any natural integer $n$ the polynomial $R_n$ as follows $$R_n(X) = (X^2 - 1)^n$$ and we now set $$L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$$
Let $n \in \mathbb{N}$.
(a) What is the degree of the polynomial $L_n$? Express $M(L_n)$ in terms of $L_n$.
(b) Show that if $n \geq 1$ then $$\forall P \in \mathbb{R}_{n-1}[X], \quad (L_n \mid P) = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq n$, we have $$L_n^{(k)}(1) = \frac{(n+k)!}{(n-k)!} \frac{1}{k! \, 2^k}$$ (d) Show that for every natural integer $k$, we have $$S\left(L_n, L_n^{(2k+1)}\right) = 2 L_n^{(2k+1)}(1)$$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The endomorphisms $T(P) = P'$ and $M(P) = P^*$ are defined as before. The polynomials $L_n$ are defined by $L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$ where $R_n(X) = (X^2-1)^n$.
Show that the pair $(L_{2m}, L_{2m-1})$ is a characterizing pair of $G$.

We introduce a uniformly distributed random variable $Z : \Omega \rightarrow \{-1,1\}^{n}$.
(a) Show that for $m \in \{0, \ldots, n-1\}$, we have $$\sum_{k=0}^{m} (n - 2k) \binom{n}{k} = n \binom{n-1}{m}.$$
(b) Deduce that for all $A \in \mathcal{M}_{n}(\{-1,1\})$, $$\mathbb{E}\left[g_{A}(Z)\right] = \frac{n^{2}}{2^{n-1}} \binom{n-1}{\left\lfloor \frac{n}{2} \right\rfloor}$$ where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor of $\frac{n}{2}$.

(a) Show that $$\underline{M}(n) \geqslant \frac{n^{2}}{2^{n-1}} \binom{n-1}{\left\lfloor \frac{n}{2} \right\rfloor}.$$
(b) Show next, using Stirling's formula recalled in the preamble, that this lower bound is equivalent to $C n^{\alpha}$ as $n$ tends to infinity, for constants $C$ and $\alpha > 0$ that one will make explicit. Compare with the upper bound for $\underline{M}(n)$ obtained in question 6 of Part II.

Determine $\mathcal{D}_{\zeta}$, the domain of definition of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

Show that $\zeta$ is continuous on $\mathcal{D}_{\zeta}$, where $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

Study the monotonicity of $\zeta$, where $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

Justify that $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$ admits a limit at $+\infty$.