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 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 $$\begin{aligned} r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\ S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Show that, if $N$ is even: $N = 2 M$ with $M \geqslant 1$, and if $0 < x \leqslant 1 / N$, we have $S _ { N } ( x ) \geqslant 0$ and $$E _ { N } ( x ) \leqslant \frac { N ! x ^ { N + 1 } } { \sum _ { \ell = 0 } ^ { M - 1 } ( 1 - ( 2 \ell + 1 ) x ) ( 2 \ell ) ! x ^ { 2 \ell + 1 } }$$

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 $$\begin{aligned} S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Verify that $E _ { 4 } \left( \frac { 1 } { 10 } \right) \leqslant 3.10 ^ { - 3 }$.

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 $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$.

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, if $f$ is a trigonometric polynomial, then $F$ is bounded on $\mathbb { R }$.

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 $t \rightarrow + \infty$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We assume $x \neq 0$ (but close to 0). We consider a new unknown function $\tilde { \alpha } : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$, of the form $$\tilde { \alpha } ( t ) = \alpha ( t ) + x h ( \alpha ( t ) ) ,$$ where $h : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is an auxiliary function, 1-periodic in each of its arguments, of class $\mathscr { C } ^ { 1 }$ and of zero average, and which moreover, for some $\nu \in \mathbb { R } ^ { 2 }$, satisfies the equation $$\forall \theta \in \mathbb { R } ^ { 2 } , \quad d h ( \theta ) \cdot \omega + g ( \theta ) = \nu . \tag{4}$$
Determine $\nu$ as a function of $g$. In the case where the two components $g _ { 1 }$ and $g _ { 2 }$ of $g$ are trigonometric polynomials, deduce the existence of a solution $h$ of equation (4), which you will make explicit.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$ We admit that $\Gamma(x) \underset{x \rightarrow +\infty}{\sim} \sqrt{2\pi} x^{(x-1/2)} \mathrm{e}^{-x}$.
Investigate whether the series on the right-hand side of $(S)$ converges absolutely when $|x| = R$, where $R$ is the radius of convergence.

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.

Prove that the sequence $(y_k)_{k \geqslant 2}$ is increasing, where $y_k$ denotes the maximum of the determinant on $\mathcal{Y}_k$.

We consider a real $\lambda$ and the sequence $\left(u_k = \lambda^k\right)_{k \in \mathbb{N}}$. What is the sequence $\left(v_k\right)_{k \in \mathbb{N}}$ defined by formula $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j \quad \text{(I.1)}$$ Then verify formula $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j \quad \text{(I.2)}$$

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 assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. Conclude that for all $g$ of $\mathscr{F}$ piecewise continuous, $$\sum_{k=0}^{+\infty} \mathbb{E}\left(g\left(x - S_k\right)\right) \rightarrow \frac{1}{\mathbb{E}(X)} \int_{-\infty}^{+\infty} g(t)\,dt \quad \text{when} \quad x \rightarrow +\infty$$

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 set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Prove the formula $$\forall n \in \mathbb { N } , \quad B _ { n + 1 } = \sum _ { k = 0 } ^ { n } \binom { n } { k } B _ { k }$$

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Show that the sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. The sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.
Deduce a lower bound for the radius of convergence $R$ of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$.

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. Let $R$ be the radius of convergence of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$, and for $x \in ] - R , R [$, set $f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { B _ { n } } { n ! } x ^ { n }$.
Show that for all $x \in ] - R , R [ , f ^ { \prime } ( x ) = \mathrm { e } ^ { x } f ( x )$.

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. Let $R$ be the radius of convergence of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$, and for $x \in ] - R , R [$, set $f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { B _ { n } } { n ! } x ^ { n }$. It has been shown that $f ^ { \prime } ( x ) = \mathrm { e } ^ { x } f ( x )$ for all $x \in ] - R , R [$.
Deduce an expression for the function $f$ on $] - R , R [$.

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 $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$
Show that the family $( H _ { 0 } , \ldots , H _ { n } )$ is a basis of the space $\mathbb { R } _ { n } [ X ]$.