In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Give the general form of sequences belonging to $\operatorname { Sol } ($ II.1 $)$ as a function of the complex roots $r _ { 1 }$ and $r _ { 2 }$ of the equation $r ^ { 2 } + a r + 1 = 0$. What are $r _ { 1 } + r _ { 2 }$ and $r _ { 1 } r _ { 2 }$?

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $| a | > 2$, the zero sequence is the only periodic solution of (II.1).

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = - 2$ then, (II.1) admits infinitely many constant solutions and infinitely many unbounded solutions.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = + 2$ then, (II.1) admits infinitely many 2-periodic solutions and infinitely many unbounded solutions.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Suppose in this question that $p$ is an integer greater than or equal to 3. Give a value of $a \in ] - 2,2 [$ for which all solutions of equation (II.1) are $p$-periodic.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2). We set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the sequence $\left( W _ { k } \right) _ { k \in \mathbb { N } }$ is constant.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by $$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$ Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form $$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$ Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that $\operatorname { det } Q = 1$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. We fix a solution $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of (II.3). Prove that, for any natural integer $k$ and any natural integer $r \in \llbracket 1 , p - 1 \rrbracket$, $$\left\{ \begin{array} { l } Z _ { k p } = Q ^ { k } Z _ { 0 } \\ Z _ { k p + r } = A _ { r - 1 } A _ { r - 2 } \cdots A _ { 0 } Q ^ { k } Z _ { 0 } \end{array} \right.$$

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that (II.2) admits a nonzero periodic solution of period $p$ if and only if 1 is an eigenvalue of $Q$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Deduce that (II.2) admits a nonzero periodic solution of period $p$ if and only if $\operatorname { tr } ( Q ) = 2$. Prove that in this case, either all solutions of (II.2) are periodic of period $p$, or (II.2) admits an unbounded solution.
One may prove that there exists a matrix $P \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ and a complex number $\alpha$ such that $Q = P \left( \begin{array} { c c } 1 & \alpha \\ 0 & 1 \end{array} \right) P ^ { - 1 }$ and, in the case where $\alpha \neq 0$, consider the solution of Sol(II.2) whose image by $\Psi$ is the vector $P \binom { 0 } { 1 }$.

Determine the limit of $\zeta(x)$ as $x$ tends to 1 from above.

Determine the limit of $\zeta(x)$ as $x$ tends to $+\infty$.

Recall the expression of the general term of the sequence $(x_k)_{k \in \mathbb{N}}$ as a function of the solutions of the equation $$bx^2 + (a-\lambda)x + c = 0 \tag{I.1}$$

Using the conditions imposed on $x_0$ and $x_{n+1}$, show that (I.1) admits two distinct solutions $r_1$ and $r_2$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Determine $\mathcal{D}_{f}$, the domain of definition of $f$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is continuous on $\mathcal{D}_{f}$ and study its variations.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Calculate $f(k)$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q12, deduce an asymptotic equivalent of $f$ at $+\infty$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Using the result of Q14, deduce an asymptotic equivalent of $f$ at $-k$. What are the right and left limits of $f$ at $-k$?

We consider the power series in the real variable $x$ given by $\sum_{k \in \mathbb{N}^{*}} (-1)^{k} \zeta(k+1) x^{k}$.
Determine the radius of convergence $R$ of this power series. Is there convergence at $x = \pm R$?

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that $f$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}_{f}$ and calculate $f^{(k)}(x)$ for all $x \in \mathcal{D}_{f}$ and all $k \in \mathbb{N}^{*}$.