Let $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x)$$

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$,
$$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$

Let $z \in D$. Show that the function $\Psi : t \mapsto ( 1 - t z ) e ^ { L ( t z ) }$ is constant on $[ 0,1 ]$, and deduce that
$$\exp ( L ( z ) ) = \frac { 1 } { 1 - z }$$

Let $z \in D$. Show that the function $\Psi : t \mapsto (1-tz)e^{L(tz)}$ is constant on $[0,1]$, and deduce that $$\exp(L(z)) = \frac{1}{1-z}$$

Show, for $r > 0$, that $$r < \rho(f) \Rightarrow \exists a > 0 \text{ such that } f \prec \frac{a}{r - z} \Rightarrow r \leqslant \rho(\hat{f})$$ deduce in particular that $\rho(\hat{f}) = \rho(f)$.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is finite for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing, and that it is constant from rank $\max ( n , 1 )$ onward.

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce that the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ is convergent for all $z$ in $D$.

Show that $|L(z)| \leq -\ln(1-|z|)$ for all $z$ in $D$. Deduce that the series $\sum_{n \geq 1} L(z^n)$ is convergent for all $z$ in $D$.

Show that $\widehat{f \cdot g} \prec \hat{f} \cdot \hat{g}$, deduce that $\rho(f \cdot g) \geqslant \min(\rho(f), \rho(g))$.

Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.

Let $f$ and $g$ be power series, with $g \in O_1$. Show that $\widehat{f \circ g} \prec \hat{f} \circ \hat{g}$. Deduce that, if $f$ and $g$ have strictly positive radius of convergence, then $\rho(f \circ g) > 0$.

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$ and $z \in \mathbf{C} \backslash \mathbb{D}$. Show that the series of matrices $\sum \frac{M^{j}}{z^{j+1}}$ converges. We will admit the following fact: let $(E, N)$ be a finite-dimensional normed vector space; if $(v_{j})_{j \in \mathbf{N}}$ is a sequence of elements of $E$ such that the series $\sum N(v_{j})$ converges, then the series $\sum v_{j}$ converges in $E$. If $m \in \mathbf{N}$, give a simplified expression for $\left(zI_{n} - M\right)\sum_{j=0}^{m} \frac{M^{j}}{z^{j+1}}$. Deduce that $$R_{z}(M) = \sum_{j=0}^{+\infty} \frac{M^{j}}{z^{j+1}}$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}.$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Calculate $b_2$, $b_4$ and $b_6$ then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Using the relation $\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = 0$ for $n \geqslant 1$, calculate $b_2$, $b_4$ and $b_6$, then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.

Let $f$ and $g$ be power series, with $g \in O_1$. For all $z$ satisfying $|z| < \rho(\hat{f} \circ \hat{g})$, show that the series $f$ converges at $g(z)$ and that $f \circ g(z) = f(g(z))$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $$\left\|f^{(k)} - P^{(k)}\right\|_{\infty} \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_{\infty}$$ for all $k \in \llbracket 0, K-1 \rrbracket$.

We fix $f \in \mathcal{C}^K([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $\left\|f^{(k)} - P^{(k)}\right\|_\infty \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_\infty$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Let $(c_{j})_{j \in \mathbf{N}}$ be a sequence of complex numbers such that the series $\sum c_{j}$ converges absolutely. We set $$\forall t \in \mathbf{R}, \quad u(t) = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$
Justify the existence and continuity of the function $u$. For $k \in \mathbf{N}$, show that $$\frac{1}{2\pi} \int_{-\pi}^{\pi} u(t) e^{i(k+1)t} \mathrm{d}t = c_{k}$$

We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
which is obviously of class $\mathcal { C } ^ { \infty }$.
Show that $\varphi _ { n , \alpha }$ and $\varphi _ { n , \alpha } ^ { \prime }$ are integrable on $] 0 , + \infty [$.

Using a series expansion under the integral, show that
$$\int _ { 0 } ^ { + \infty } \ln \left( 1 - e ^ { - u } \right) \mathrm { d } u = - \frac { \pi ^ { 2 } } { 6 }$$

Using a series expansion under the integral, show that $$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$

Let $f, g$ and $h$ be power series, with $g, h \in O_1$, show that $(f \circ g) \circ h = f \circ (g \circ h)$.

Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^{K}([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_{\infty} \leqslant \left\|f^{(K)}\right\|_{\infty} + C \sum_{\ell=1}^{K} \left|f\left(x_{\ell}\right)\right|$$ is satisfied.