We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$, and that two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$ with strictly positive radius of convergence, $b_1 > 0$, $c_1 < 0$, satisfy $\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 }$ for $q$ near 0 in $[0,+\infty[$.
Calculate $b _ { 1 } , b _ { 2 } , b _ { 3 }$ and $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$. Deduce the following asymptotic expansions when $q \rightarrow 0 , q > 0$, for the functions $\phi _ { - } ^ { - 1 }$ and $\phi _ { + } ^ { - 1 }$ as well as their derivatives: $$\begin{array} { l l } \phi _ { + } ^ { - 1 } ( q ) = \sqrt { 2 q } + \frac { 2 q } { 3 } + \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) , & \phi _ { - } ^ { - 1 } ( q ) = - \sqrt { 2 q } + \frac { 2 q } { 3 } - \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) \\ \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) = \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } + \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) , & \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) = - \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } - \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) \end{array}$$

We recall that $\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { - 1 } ^ { + \infty } e ^ { - y \phi ( s ) } d s$ where $\phi ( s ) = s - \ln(1+s)$, and that $\phi_-^{-1} : ]0,+\infty[ \to ]-1,0[$ and $\phi_+^{-1} : ]0,+\infty[ \to ]0,+\infty[$ are the inverse bijections of the restrictions of $\phi$.
Show that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { 0 } ^ { \infty } e ^ { - y q } \left( \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) - \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) \right) d q$$

Using the results of the previous questions, deduce that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \left( \frac { 2 \pi } { y } \right) ^ { 1 / 2 } \left( 1 + \frac { 1 } { 12 y } + o \left( \frac { 1 } { y } \right) \right) \quad \text { when } y \rightarrow + \infty .$$

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Simplify as much as possible the following expressions: $$\sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^n \quad \text{and} \quad \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^{n+1}$$

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: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ 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.

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

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.

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 $r$.

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 $+\infty$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
We denote $$g_{j} = \int_{-1}^{1} P_{j}(x)^{2}\,dx, \quad I_{j} = \int_{-1}^{1} \left(1 - x^{2}\right)^{j}\,dx$$
(a) Establish a relation between $g_{j}$ and $I_{j}$.
(b) Find a relation between $I_{j}$ and $I_{j-1} - I_{j}$, and deduce a recurrence relation for the sequence $\left(I_{j}\right)_{j \in \mathbb{N}}$.
(c) Deduce the value of $I_{j}$, then that of $g_{j}$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. The subspace of $\mathbb{R}_{n}[X]$ formed by even polynomials is denoted $\Pi_{n}$, and that of odd polynomials is denoted $J_{n}$.
(a) Show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is a basis of $\mathbb{R}_{n}[X]$.
(b) Deduce that the family $\left(P_{2j}\right)_{0 \leqslant j \leqslant \frac{n}{2}}$ is a basis of $\Pi_{n}$, while the family $\left(P_{2j+1}\right)_{0 \leqslant j \leqslant \frac{n-1}{2}}$ is a basis of $J_{n}$.

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We assume in this question that $U_{N}$ is even; we thus have $U_{N} \in \Pi_{n}$. In $\Pi_{n}$, the equation $P(1) = 1$ defines an affine subspace denoted $H_{n}$.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
(a) Show that $$\left\|U_{N}\right\|_{2} = \min\left\{\|P\|_{2} \mid P \in H_{n}\right\}$$
(b) Deduce that there exists a real number $\mu$ such that for all integers $0 \leqslant j \leqslant \frac{n}{2}$, we have $\left\langle U_{N}, P_{2j} \right\rangle = \mu$. (One may consider polynomials $P \in H_{n}$ of the form $U_{N} + t\left(P_{2j} - P_{2k}\right)$ with $t \in \mathbb{R}$.)
(c) Express $U_{N}$ in the basis of $P_{2j}$. Deduce that $$\frac{1}{\mu} = \sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}$$
(d) Establish in this case the formula $$a_{N} = \left(\sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}\right)^{-1}.$$

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Demonstrate, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { q = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { q } } { q ! } x ^ { q } ( 1 - x ) ^ { - q / 2 }$$