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 $p$ ?

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that the application which associates to $f$ the function $Lf$ is injective.

We assume that $f$ is positive and that $E$ is neither empty nor equal to $\mathbb{R}$. We denote by $\alpha$ its infimum.
VII.A.1) Show that if $Lf$ is bounded on $E$, then $\alpha \in E$.
VII.A.2) If $\alpha \notin E$, what can we say about $Lf(x)$ when $x$ tends to $\alpha^+$?

In this question, $f(t) = \cos t$ and $\lambda(t) = \ln(1+t)$.
VII.B.1) Determine $E$.
VII.B.2) Determine $E^{\prime}$.
VII.B.3) Show that $Lf$ admits a limit at $\alpha$, the infimum of $E$, and determine it.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients and we use the transformation $L$ applied to elements of $\mathcal{P}$ for the study of an operator $U$.
Let $P$ and $Q$ be two elements of $\mathcal{P}$.
Show that the integral $\displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, where $\bar{P}$ is the polynomial whose coefficients are the conjugates of those of $P$, converges.

Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.
We recall that this implies two things:

• on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
• on the other hand, the elements of $J_x$ are the multiples of $B$.

By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.

In $\mathbb{K}^{\mathbb{N}}$, what are the linear recurrent sequences of order 0? of order 1?
What are the sequences in $\mathbb{K}^{\mathbb{N}}$ whose minimal polynomial is $(X-1)^2$?

We consider the sequence $x$ defined by $x_0 = 0, x_1 = -1, x_2 = 2$ and by the linear recurrence relation of order 3: $\forall n \in \mathbb{N}, x_{n+3} = -3x_{n+2} - 3x_{n+1} - x_n$.
Determine the minimal polynomial (and thus the minimal order) of the sequence $x$.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Prove that $\mathcal{R}_A(\mathbb{K})$ is a vector subspace of dimension $p$ of $\mathbb{K}^{\mathbb{N}}$ and that it is stable under $\sigma$ (we do not ask here to determine a basis of $\mathcal{R}_A(\mathbb{K})$, as this is the object of the following questions).

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Determine $\mathcal{R}_A(\mathbb{K})$ when $A = X^p$ (with $p \geqslant 1$) and give a basis for it.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$.
We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$.
a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension.
b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.

In this question, we assume that the polynomial $A$ is split over $\mathbb{K}$. More precisely, we write $A = X^{m_0} \prod_{k=1}^{d} (X - \lambda_k)^{m_k}$, where:

• the scalars $\lambda_1, \lambda_2, \ldots, \lambda_d$ are the distinct non-zero roots of $A$ in $\mathbb{K}$, and $m_1, m_2, \ldots, m_d$ are their respective multiplicities (greater than or equal to 1). If $A$ has no non-zero root, we adopt the convention that $d = 0$ and that $\prod_{k=1}^{d}(X-\lambda_k)^{m_k} = 1$;
• the integer $m_0$ is the multiplicity of 0 as a possible root of $A$. If 0 is not a root of $A$, we adopt the convention $m_0 = 0$.

With these notations, we have $\sum_{k=0}^{d} m_k = \deg A = p$.
Using the kernel decomposition theorem, show that $\mathcal{R}_A(\mathbb{K})$ is the set of sequences $x = (x_n)_{n \geqslant 0}$ in $\mathbb{K}^{\mathbb{N}}$ such that: $$\forall n \geqslant m_0, \quad x_n = \sum_{k=1}^{d} Q_k(n) \lambda_k^n$$ where, for all $k$ in $\{1, \ldots, d\}$, $Q_k$ is in $\mathbb{K}[X]$ with $\deg Q_k < m_k$.
Remark: if $d = 0$, the sum $\sum_{k=1}^{d} Q_k(n)\lambda_k^n$ is by convention equal to 0.

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$.
Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that if $n \geqslant p$, the map $\varphi_n : \left\{ \begin{array}{l} \mathcal{R}_B(\mathbb{K}) \rightarrow \mathbb{K}^n \\ v \mapsto (v_0, \ldots, v_{n-1}) \end{array} \right.$ is injective.
Deduce from this that if $n \geqslant p$, then $\operatorname{rang}(H_n(x)) = p$.
Remark: it is clear that this result remains true if $p = 0$ (since the sequence $x$ and the matrices $H_n(x)$ are zero).

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. The kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
With these notations, show that the minimal polynomial of $x$ is $B = X^p + b_{p-1}X^{p-1} + \cdots + b_1 X + b_0$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
In the computer language of your choice (which you will specify), write a procedure (or function) with parameter a natural number $n$ and returning the list (or sequence, or vector) of $x_k$ for $0 \leqslant k \leqslant n$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Specify the rank of $H_n(x)$ for any integer $n$ in $\mathbb{N}^*$ and indicate the minimal order of the sequence $x$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Determine the minimal recurrence relation of the sequence $x$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Give a formula allowing for any $n \geqslant 1$ to directly compute $x_n$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
We decide to modify only the value of $x_0$, by setting this time $x_0 = \frac{1}{2}$.
With this modification, quickly redo the study of questions II.C.2 and II.C.3.

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 $n \in \mathbb{N}$.
Let $P$ be a polynomial with real coefficients. Show that $$( 1 - x ) \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } P \left( x ^ { n } \right) \underset { \substack { x \rightarrow 1 \\ x < 1 } } { \longrightarrow } \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t$$ We will first consider the special case $P ( x ) = x ^ { k }$, where $k \in \mathbb { N }$.

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 $n \in \mathbb{N}$. Let $g$, $g^+$, $g^-$, $P$, $Q$ be as defined in II.E.1--II.E.3. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$.
Establish the existence of an integer $N _ { 1 } > 0$ such that for every integer $N \geqslant N _ { 1 }$, $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } P \left( x _ { N } ^ { n } \right) \geqslant \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t - \varepsilon$$ and $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } Q \left( x _ { N } ^ { n } \right) \leqslant \int _ { 0 } ^ { 1 } Q ( t ) \mathrm { d } t + \varepsilon$$

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 $A_N = \sum_{k=0}^N a_k$. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Let $N_1$ be as in II.E.4.
Deduce from the three previous questions that for every integer $N \geqslant N _ { 1 }$ $$1 - 5 \varepsilon \leqslant \left( 1 - x _ { N } \right) A _ { N } \leqslant 1 + 5 \varepsilon$$

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 (i.e., prove property II.3: $\lim_{n \to \infty} \widetilde{a}_n = 1$).