Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. From the previous questions, there exist $a_n \in \mathbb{C}$ such that $c_{n,f}(r) = a_n r^{|n|}$ for all $r \in \mathbb{R}_+^*$.
By stating precisely the theorem used, establish $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = \lim _ { p \rightarrow \infty } \sum _ { n = - p } ^ { p } a _ { n } r ^ { | n | } e ^ { i n \theta }$$

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that this function series converges pointwise on $\mathbb { R }$.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that $p$ is well defined, continuous and $2 \pi$-periodic.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that the function $p$ is not of class $\mathcal { C } ^ { 1 }$.

Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$.
Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy: $$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$

Let $f \in \mathcal{C}$. If $f$ is differentiable at $x_{0}$, show that $f \in \Gamma^{s}(x_{0})$ for all $s \in [0,1[$.

Show that for all $x_{0} \in ]0,1[$, there exists $f \in \mathcal{C}$ non-differentiable at $x_{0}$ such that for all $s \in [0,1[$, $f \in \Gamma^{s}(x_{0})$.

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Using the result of question 3b, deduce that $\omega_{f}$ is continuous on $[0,1]$.

We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We denote $g$ the function defined on $[ - 1 , + \infty [$ by $$g ( x ) = \sum _ { n = 2 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$$
Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $[ - 1 , + \infty [$.
Specify in particular the value of $g ^ { ( k ) } ( 0 )$ as a function of $\zeta ( k + 1 )$ for every integer $k \geqslant 1$.

We consider the space $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { d } \right)$ of functions $f : \mathbb { R } ^ { d } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of their variables, equipped with the uniform norm $\| f \| _ { \infty } = \sup _ { \theta \in \mathbb { R } ^ { d } } | f ( \theta ) |$. A trigonometric polynomial (in $d$ variables) is any function of the form $\theta \mapsto \sum _ { k \in K } c _ { k } e ^ { 2 i \pi k \cdot \theta }$ where $K$ is a finite subset of $\mathbb { Z } ^ { d }$. We work in dimension $d = 2$. The subspace $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ of $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ consists of functions of the form $\theta = \left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \sum _ { i = 1 } ^ { n } f _ { i } \left( \theta _ { 1 } \right) g _ { i } \left( \theta _ { 2 } \right)$, where $n \in \mathbb { N } ^ { * }$ and $f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$. We admit that trigonometric polynomials in one variable are dense in $\mathscr{C}_{\text{per}}(\mathbb{R})$.
Show that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$.

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $$\forall t \in ] - 1 / 2,1 / 2 ] , \quad \psi _ { j } ( t ) = \max ( 0,1 - j | t | ) .$$ For integers $0 \leqslant k < j$, the functions $\psi _ { j , k } : \mathbb { R } \rightarrow \mathbb { R }$ are defined by $$\forall t \in \mathbb { R } , \quad \psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right) .$$
Show that $\psi _ { j , k } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$.

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Let $j \geqslant 2 , k _ { 1 }$ and $k _ { 2 }$ be two integers such that $0 \leqslant k _ { 1 } , k _ { 2 } < j$, and $$\theta \in \left[ \frac { k _ { 1 } } { j } , \frac { k _ { 1 } + 1 } { j } \left[ \times \left[ \frac { k _ { 2 } } { j } , \frac { k _ { 2 } + 1 } { j } [ . \right. \right. \right.$$
Express $S _ { j } ( f ) ( \theta )$ as a barycenter of the points $f \left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ where $\ell _ { 1 } \in \left\{ k _ { 1 } , k _ { 1 } + 1 \right\}$ and $\ell _ { 2 } \in \left\{ k _ { 2 } , k _ { 2 } + 1 \right\}$. Deduce that $\left\| S _ { j } ( f ) - f \right\| _ { \infty } \rightarrow 0$ when $j \rightarrow + \infty$.

Using the results of questions 8, 9, 10a and 10b, conclude that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$.

In the formula $f(x) = \sum_{i=0}^{+\infty} q_i g\left(x - y_i\right)$, show that the convergence of the series is normal on every segment of $\mathbb{R}$. One may use question 3c.

Suppose that $g$ is continuous. Show that $f$ is uniformly continuous.

Suppose that $g$ is of class $\mathscr{C}^1$. Show that $g'$ is bounded. Deduce that $f$ is of class $\mathscr{C}^1$, that $f'$ is bounded and uniformly continuous and that for all $x \in \mathbb{R}$, $$f'(x) = g'(x) + \sum_{i=0}^{+\infty} p_i f'\left(x - x_i\right)$$

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 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.

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 $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

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 $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ 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 $\mathcal { D }$ 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 $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p }$.

For every $s > 1$, let $\zeta(s) = \sum_{n=1}^{+\infty} \frac{1}{n^s}$. Show that $\zeta$ is continuous on $]1, +\infty[$.