We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$. We set, for $n \in \mathbb{N}$, $$R_n(j) = \sum_{p=n+1}^{+\infty} p^j \alpha_p$$
Show that the sequence $(R_n(j))_{n \in \mathbb{N}}$ has rapid decay.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$ For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Show that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ The Fourier coefficients of $\widetilde{h}$ are given by: $$a_0(\widetilde{h}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \widetilde{h}(t)\, dt, \quad a_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \cos(nt)\, dt, \quad b_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \sin(nt)\, dt.$$
Let $f \in C^\infty([-1,1])$.
Show that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ has rapid decay. What is the value of $b_n(\widetilde{f})$?

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
Show that we can construct a sequence $(p_n)_{n \in \mathbb{N}}$ of polynomial functions such that:

• for every integer $n$, $\deg(p_n) \leqslant n$;
• $(\|f - p_n\|_\infty)_{n \in \mathbb{N}}$ has rapid decay.

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Using the inequality from question I.A.1, show that $R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + O \left( \frac { 1 } { n ^ { \alpha } } \right)$.

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Deduce that $$R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + \frac { 1 } { 2 n ^ { \alpha } } + O \left( \frac { 1 } { n ^ { \alpha + 1 } } \right)$$

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$.
Deduce the asymptotic expansion of the remainder $$R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } } = - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + a _ { 2 } f ^ { \prime \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right) + O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$$

Give the asymptotic expansion of $R _ { n } ( 3 )$ corresponding to the case $\alpha = 3$ and $p = 3$.

Show that, in the expression of $R _ { n } ( \alpha )$ from II.B.2, the term $O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$ can be written in the form of an integral.

What can be said about the approximation of $S ( \alpha )$ by $\widetilde { S } _ { n , 2 p } ( \alpha )$ when, with $n$ fixed, $p$ tends to $+ \infty$ ? For the numerical calculation of $S ( \alpha )$, how should one choose $n$ and $p$ ?

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ and $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
Deduce that $$\left| v_{n} - \frac{1}{12n} \right| \leqslant \frac{1}{12n^{2}}$$ then that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + \frac{1}{12n} + O\left(\frac{1}{n^{2}}\right)$$

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$: $$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$
IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion: $$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$
IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion: $$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$ where $A_n = \sum_{k=0}^n a_k$.
Conclude that there exists a real $\nu > 0$ such that from a certain rank onwards we have $\widetilde { a } _ { n } \geqslant \nu$.

Show that for all $k$ in $\mathbb { N } ^ { * } , \left| \varphi _ { n } ( x ) \right| = o \left( \frac { 1 } { n ^ { k } } \right)$ as $n$ tends to $+ \infty$.
Use Fourier series of successive derivatives of $G _ { x }$.

In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $M_n = \sum_{k=1}^n \dfrac{1}{2\cos\dfrac{k\pi}{2n+1}}$.
Give an equivalent of $M_n$ as $n$ tends to $+\infty$.

Let $q : [0,1] \rightarrow \mathbf{R}$ defined by $$\left\{ \begin{array}{l} q(x) = x \cos\left(\frac{\pi}{x}\right) \text{ for } x > 0 \\ q(0) = 0 \end{array} \right.$$ Show that for all $x_{0} \in [0,1]$, $\alpha_{q}(x_{0}) = 1$, but that $\frac{\omega_{q}(h)}{\sqrt{h}}$ does not tend to 0 when $h$ tends to 0.

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 $n_{0} < n_{1}$.

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 $H _ { n } \sim \ln n$.

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 $N \geqslant 1$, $$R _ { N } ( x ) \sim r _ { N } ( x ) \quad \text { when } \quad x \rightarrow 0$$

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 $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }$$
Show that, for $0 < x < 1 / 2$, the sequence $\left( \left| r _ { N } ( x ) \right| \right) _ { N \geqslant 1 }$ is decreasing up to a certain rank, then increasing.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that more generally, if $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$, then $F ( t ) = o ( t )$ when $t \rightarrow + \infty$.

If $\lambda > 1$, and $C_{n} = \sum_{k=0}^{n} \frac{(n\lambda)^{k}}{k!}$, determine an equivalent of $C_{n}$ when $n \rightarrow +\infty$.
Consider the integral $\frac{1}{n!} \int_{-\infty}^{0} (r - t)^{n} \mathrm{e}^{t} \mathrm{~d}t$ and choose the real number $r$ appropriately.

Let $\ell > 0$ be fixed. Determine the behaviour of $\mathbb{E}(N(x, x+\ell))$ when $x \rightarrow +\infty$. Interpret the result. Is this result true if there exists $d > 0$ such that $\mathbb{P}(X \in d\mathbb{Z}) = 1$?

We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$
Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.

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.