Let $x \in \mathcal{D}_{\zeta}$ and let $n \in \mathbb{N}$ such that $n \geqslant 2$. Show: $$\int_{n}^{n+1} \frac{\mathrm{~d}t}{t^{x}} \leqslant \frac{1}{n^{x}} \leqslant \int_{n-1}^{n} \frac{\mathrm{~d}t}{t^{x}}$$

Using the result of Q5, deduce that for all $x \in \mathcal{D}_{\zeta}$, $$1 + \frac{1}{(x-1)2^{x-1}} \leqslant \zeta(x) \leqslant 1 + \frac{1}{x-1}$$

Determine the limit of $\zeta(x)$ as $x$ tends to 1 from above.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Determine $\mathcal{D}_{f}$, the domain of definition of $f$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is continuous on $\mathcal{D}_{f}$ and study its variations.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Calculate $f(k)$.

We consider the power series in the real variable $x$ given by $\sum_{k \in \mathbb{N}^{*}} (-1)^{k} \zeta(k+1) x^{k}$.
Determine the radius of convergence $R$ of this power series. Is there convergence at $x = \pm R$?

We denote by $n$ the integer part of $\frac{N}{2}$. 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 $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Discuss, depending on the parity of $n$, the value of $a_{N}$. We will give its explicit value.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}_{f}$ and calculate $f^{(k)}(x)$ for all $x \in \mathcal{D}_{f}$ and all $k \in \mathbb{N}^{*}$.

We denote by $n$ the integer part of $\frac{N}{2}$. For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$.
Give the explicit formula for $R_{N}$, in terms of the polynomials $P_{j}$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that there exists $A \in \mathbb{R}_{+}^{*}$ such that $$\forall k \in \mathbb{N}^{*}, \forall x \in {]-1,1[}, \quad \left|f^{(k)}(x)\right| \leqslant k! \left(A + \frac{1}{(x+1)^{k+1}}\right)$$

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q18, deduce that $f$ is expandable as a power series on $]-1,1[$ and that $$\forall x \in {]-1,1[}, \quad f(x) = \sum_{k=1}^{+\infty} (-1)^{k} \zeta(k+1) x^{k}$$

We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 [$ and all $q \in \mathbb { N } ^ { * }$, we have $$( 1 - x ) ^ { - q / 2 } = \sum _ { p = 0 } ^ { + \infty } H _ { p } \left( \frac { q } { 2 } + p - 1 \right) x ^ { p }$$

We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Deduce $$\forall x \in ] - 1,1 \left[ , \quad \varphi ( x ) = 1 + \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } a _ { i , j } ( x ) \right) \right.$$ where we have set $$\forall ( i , j ) \in \mathbb { N } ^ { 2 } , \quad a _ { i , j } ( x ) = \frac { ( - 1 ) ^ { i + 1 } } { ( i + 1 ) ! } H _ { j } \left( \frac { i - 1 } { 2 } + j \right) x ^ { i + j + 1 }$$

Using the results of the previous questions, deduce an integral expression of $\zeta(k+1)$ for all $k \in \mathbb{N}^{*}$.

We assume $c(x) = 0$ for all $x \in [0,1]$. Let $u$ be the solution of problem (1) and $u_0, \ldots, u_n$ solutions of system (2) with $c = 0$. Define: $$\hat { B } _ { n + 1 } u ( X ) = \sum _ { k = 0 } ^ { n } u _ { k } \binom { n + 1 } { k } X ^ { k } ( 1 - X ) ^ { n + 1 - k }$$
Show that for all $n \in \mathbb { N } ^ { * }$ and all $x \in ]0,1[$ we have:
$$\left( \hat { B } _ { n + 1 } u \right) ^ { \prime \prime } ( x ) = - \frac { n } { n + 1 } \sum _ { \ell = 0 } ^ { n - 1 } f \left( \frac { \ell + 1 } { n + 1 } \right) \binom { n - 1 } { \ell } x ^ { \ell } ( 1 - x ) ^ { n - 1 - \ell }$$

We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 \left[ , \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } \left| a _ { i , j } ( x ) \right| \right) = \exp \left( \frac { | x | } { \sqrt { 1 - | x | } } \right) - 1 \right.$.

Show that $$\forall k \in \mathbb{N}^{*}, \quad \zeta(k+1) = \frac{1}{k!} \int_{0}^{+\infty} \frac{u^{k}}{\mathrm{e}^{u} - 1} \mathrm{~d}u$$

We assume $c(x) = 0$ for all $x \in [0,1]$, $f \in \mathcal{C}([0,1],\mathbb{R})$ satisfies $|f(y)-f(z)| \leq K|y-z|^\alpha$ for some $\alpha \in ]0,1]$ and $K \geq 0$. Let $u$ be the solution of problem (1) and define $\hat{B}_{n+1}u$ as above. Let $n \in \mathbb { N } ^ { * }$ such that $n \geq 2$. We set $\chi _ { n + 1 } = \hat { B } _ { n + 1 } u - u$.
(a) Show that
$$\left\| \chi _ { n + 1 } ^ { \prime \prime } \right\| _ { \infty } \leq \left\| f - B _ { n - 1 } f \right\| _ { \infty } + \frac { 1 } { n + 1 } \| f \| _ { \infty } + K \frac { 1 } { ( n + 1 ) ^ { \alpha } }$$
(b) Show that for all $x \in [ 0,1 ]$ there exists $\xi \in [ 0,1 ]$ such that
$$\chi _ { n + 1 } ( x ) = - \frac { 1 } { 2 } x ( 1 - x ) \chi _ { n + 1 } ^ { \prime \prime } ( \xi )$$
Hint: for $x \in ]0,1[$ one may consider the function
$$h ( t ) = \chi _ { n + 1 } ( t ) - \frac { \chi _ { n + 1 } ( x ) } { x ( 1 - x ) } t ( 1 - t ) , \quad t \in [ 0,1 ]$$

We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Use the results admitted in the preamble to establish the equality $$\forall x \in ] - 1,1 \left[ , \quad \varphi ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$$ where $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$

Deduce that there exists a constant $M \geq 0$ such that for all $n \in \mathbb { N } ^ { * }$, we have
$$\left\| u - \hat { B } _ { n + 1 } u \right\| _ { \infty } \leq \frac { M } { n ^ { \alpha / 2 } }$$

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that $$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } a _ { n }$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$.
Let $p \in \mathbb { N } ^ { * }$. Show that the power series $\sum _ { n \geqslant 0 } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } x ^ { n }$ converges normally on $[ 0,1 ]$ and give the value of $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p }$.

The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that all moments of order $p$ of the sequence $(a _ { n })$ are zero.