Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$. We suppose that $\|f\|_{\infty} = 1$.
Show that there exists a unique $n_{1} \in \mathbf{N}$ such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that for all $n \geq n_{1}$, we have $$\|f - S_{n} f\|_{\infty} \leq 2^{s+1} |x - x_{0}|^{s}$$ One may use the results of questions 9a and 9c.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that when $n_{0} < n_{1}$, we have $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{1} 3^{s} (n_{1} - n_{0}) |x - x_{0}|^{s} .$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We suppose furthermore that the function $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists a real number $c_{4}(N) > 0$, such that for all $h \in ]0,1]$, $$\omega_{f}(h) \leq c_{4}(N) (1 + |\log_{2} h|)^{-N}$$ For all integer $N \geq 1$, we set $c_{5}(N) = 3^{s} c_{1} (c_{4}(N))^{1/N}$. Show that $$n_{1} - n_{0} \leq n_{1} + 1 \leq \left(\frac{c_{4}(N)}{\omega_{f}(2^{-n_{1}})}\right)^{\frac{1}{N}}$$ and deduce $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{5}(N) |x - x_{0}|^{(1 - \frac{1}{N})s}$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We suppose furthermore that $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists $c_{4}(N) > 0$ such that for all $h \in ]0,1]$, $\omega_{f}(h) \leq c_{4}(N)(1 + |\log_{2} h|)^{-N}$.
Deduce from the above that $\alpha_{f}(x_{0}) \geq s$.
One may distinguish the cases $n_{0} \geq n_{1}$ and $n_{0} < n_{1}$.

We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$.
Let $\mathcal{B} = \left\{A \in \mathcal{M}_d(\mathbb{R}), \|A\| < R\right\}$. Show that the map $\varphi : A \mapsto \varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$ is defined and continuous on $\mathcal{B}$.

We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$, and $\varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$.
Let $A \in \mathcal{M}_d(\mathbb{R})$ be a nonzero matrix such that $\|A\| < R$.
1) Establish the existence of an integer $r \in \mathbb{N}^*$ such that the family $\left(A^k\right)_{0 \leqslant k \leqslant r-1}$ is free and the family $\left(A^k\right)_{0 \leqslant k \leqslant r}$ is dependent.
2) For $n \in \mathbb{N}$, show the existence and uniqueness of an $r$-tuple $(\lambda_{0,n}, \ldots, \lambda_{r-1,n})$ in $\mathbb{R}^r$ such that $$A^n = \sum_{k=0}^{r-1} \lambda_{k,n} A^k$$
3) Show that there exists a constant $C > 0$ such that: $$\forall n \in \mathbb{N}, \quad \sum_{k=0}^{r-1} |\lambda_{k,n}| \leqslant C \left\|A^n\right\|$$
4) Deduce that, for every integer $k$ between 0 and $(r-1)$, the series $\sum_{n \geqslant 0} a_n \lambda_{k,n}$ is absolutely convergent in $\mathbb{C}$.
5) Conclude that there exists a unique polynomial $P \in \mathbb{R}[X]$ such that $\varphi(A) = P(A)$ and $\deg P < r$.
6) Determine this polynomial $P$ when $A = \begin{pmatrix} 0 & -1 & -1 \\ -1 & 0 & -1 \\ 1 & 1 & 2 \end{pmatrix}$ and $a_n = \frac{1}{n!}$ for all $n \in \mathbb{N}$.

We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$, and $\varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$ defined on $\mathcal{B} = \{A \in \mathcal{M}_d(\mathbb{R}), \|A\| < R\}$.
Find a necessary and sufficient condition on the power series $\sum_{n \geqslant 0} a_n z^n$ for there to exist $P \in \mathbb{R}[X]$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad \varphi(A) = P(A)$$

State the theorem allowing the product of two series of complex numbers. (We admit in the rest of Part III that the result valid for series of complex numbers still holds for series of matrices in $\mathcal{M}_d(\mathbb{C})$.)

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Let $t \in [0, 2\pi]$ be fixed. Determine two complex numbers $\alpha$ and $\beta$, independent of $t$ and $z$, such that $$\mathrm{N}(x,y,t) = -1 + \frac{\alpha}{1 - ze^{-it}} + \frac{\beta}{1 - \bar{z}e^{it}}$$

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0. We define $u_n$, $u_n^{(r)}$, $U(s)=\sum_{n=1}^{+\infty}u_n s^n$ and $U_r(s)=\sum_{n=1}^{+\infty}u_n^{(r)}s^n$ for $s\in[-1,1]$.
Deduce that, for every strictly positive integer $r$, $U_r=U^r$ ($U^r$ denotes $U\times U\times\cdots\times U$ $r$ times).

Justify that the series with general term $a _ { n } = \frac { 1 } { n } - \int _ { n - 1 } ^ { n } \frac { \mathrm {~d} t } { t }$ converges.

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Show that there exists a real constant $A$ such that $H _ { n } \underset { + \infty } { = } \ln n + A + o ( 1 )$. Deduce that $H _ { n } \sim \ln n$.

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$. We denote $\zeta$ the function defined for $x > 1$ by $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$.
Let $r$ be a natural integer. For which values of $r$ is the series $\sum _ { n \geqslant 1 } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ convergent?
In the rest of the problem we will denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.

Give without proof the power series expansions of the functions $t \mapsto \ln ( 1 - t )$ and $t \mapsto \frac { 1 } { 1 - t }$ as well as their radius of convergence.

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$.
Deduce that the function $$t \mapsto - \frac { \ln ( 1 - t ) } { 1 - t }$$ is expandable as a power series on $] - 1,1 [$ and specify its power series expansion using the real numbers $H _ { n }$.

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 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 for every integer $n$ and for every $x$ in $] - 1,1 [$ $$\left| g ( x ) - \sum _ { k = 0 } ^ { n } \frac { g ^ { ( k ) } ( 0 ) } { k ! } x ^ { k } \right| \leqslant \zeta ( 2 ) | x | ^ { n + 1 }$$
Show that $g$ is expandable as a power series on $] - 1,1 [$.

We denote $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$. We have shown that for every real $x > -1$, $\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + x } \right)$.
Prove that for every $x$ in $] - 1,1 [$, $$\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } ( - 1 ) ^ { n + 1 } \zeta ( n + 1 ) x ^ { n }$$

We define $\varphi$ the function defined on $] - 1 , + \infty [$ by $\varphi ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$.
Show that $\varphi$ is $\mathcal { C } ^ { \infty }$ on its domain of definition and give for every natural integer $n \geqslant 2$ the value of $\varphi ^ { ( n ) } ( 0 )$ as a function of the successive derivatives of $\psi$ at the point 1.

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $\varphi ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$. We have $x B(x) = \varphi(x)$ for $x > 0$, and $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$. We have also shown that $\psi ( 1 + x ) = \psi ( 1 ) + \sum _ { n = 1 } ^ { + \infty } ( - 1 ) ^ { n + 1 } \zeta ( n + 1 ) x ^ { n }$ for $x \in ] -1, 1[$.
Conclude that, for every integer $r \geqslant 3$, $$2 S _ { r } = r \zeta ( r + 1 ) - \sum _ { k = 1 } ^ { r - 2 } \zeta ( k + 1 ) \zeta ( r - k )$$

We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Show that if $s \in ] - 1,1 [$, $$\phi ( s ) = s ^ { 2 } \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k + 2 } s ^ { k }$$
We admit the existence of two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$, in the variable $q$, and of strictly positive radius of convergence, where $b _ { 1 } > 0$ and $c _ { 1 } < 0$, and such that we have, for $q$ in a neighborhood of 0 in $[ 0 , + \infty [$, $$\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 } .$$

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Specify the domain of convergence of the series $\sum _ { k \geqslant 1 } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k }$ and show that the sequence $\left( R _ { N } ( x ) \right) _ { N \geqslant 1 }$ is not bounded.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that, if $N \in \mathbb { N } ^ { * }$ and $x > 0$, $$\left| R _ { N } ( x ) \right| \leqslant \left| r _ { N } ( x ) \right|$$ Deduce that $R _ { N + 1 } ( x ) = o \left( r _ { N } ( x ) \right)$ when $x \rightarrow 0$.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that the remainder is of the order of the first neglected term, that is, for all $N \geqslant 1$, $$R _ { N } ( x ) \sim r _ { N } ( x ) \quad \text { when } \quad x \rightarrow 0$$