For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Show that for fixed $x \in \mathbb{R}$, the sequence $(P_p(x))_{p \in \mathbb{N}^*}$ satisfies a linear relation of order 2, which we shall specify.

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Let $x \in \mathbb{R}$ such that $|2 - x| < 2$. After justifying the existence of a unique $\theta \in ]0, \pi[$ such that $2 - x = 2\cos\theta$, determine $P_p(x)$ as a function of $\sin((p+1)\theta)$ and $\sin(\theta)$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded.
Deduce that the series $\sum a _ { n } b _ { n }$ converges.

Show that, for all $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$, the series $\sum _ { n \geq 1 } \frac { e ^ { \mathrm { i } n \theta } } { n }$ converges.

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that this function series converges pointwise on $\mathbb { R }$.

We consider the function series $\sum _ { n \geq 1 } \frac { \sin ( n x ) } { \sqrt { n } }$, where $x$ is a real variable.
Show that it cannot be the Fourier series of a $2 \pi$-periodic piecewise continuous function.
One may begin by recalling Parseval's formula.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that $p$ is well defined, continuous and $2 \pi$-periodic.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Determine the Fourier series of $p$.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Show that the function $p$ is not of class $\mathcal { C } ^ { 1 }$.

Let $\theta \in \mathbb { R }$.
Determine the radius of convergence of the power series $\sum \frac { e ^ { \mathrm { i n } \theta } } { n } x ^ { n }$.

Let $\theta \in \mathbb { R }$. Let $g$ be the function from $] - 1,1 [$ to $\mathbb { C }$ defined by $$g ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } n \theta } } { n } x ^ { n }$$
a) Show that $g$ is of class $C ^ { 1 }$ on $] - 1,1 [$ and that, for all $x \in ] - 1,1 [$, $$g ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { \mathrm { i } \theta } - x } { x ^ { 2 } - 2 x \cos \theta + 1 }$$
b) Show that, if $x \in ] - 1,1 [$, $$h ( x ) = - \frac { 1 } { 2 } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) + \mathrm { i } \arctan \left( \frac { x \sin \theta } { 1 - x \cos \theta } \right)$$ is well defined and that $h ( x ) = g ( x )$.

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, $$\sum _ { k = 1 } ^ { n } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { \mathrm { i } \theta } \frac { 1 - \left( \mathrm { e } ^ { \mathrm { i } \theta } t \right) ^ { n } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Deduce that $$\sum _ { k = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { \mathrm { i } \theta } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$
One may use the dominated convergence theorem.

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Deduce that $$\sum _ { k = 1 } ^ { + \infty } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = - \frac { 1 } { 2 } \ln ( 2 - 2 \cos \theta ) + \mathrm { i } \arctan \left( \frac { \sin \theta } { 1 - \cos \theta } \right)$$

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Justify the existence of $s$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Let $x \in [ 0,1 ]$ and $n \in \mathbb { N } ^ { * }$. Show $$s ( x ) - s _ { n } ( x ) = r _ { n } x ^ { n + 1 } - \sum _ { k = n + 1 } ^ { + \infty } r _ { k } \left( x ^ { k } - x ^ { k + 1 } \right)$$

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Show that $s$ is continuous on $[ 0,1 ]$.
For continuity at 1, fix $\varepsilon > 0$ and show that if the natural integer $N$ satisfies $\left| r _ { n } \right| \leqslant \varepsilon$ for all $n \geqslant N$, then $\left| s ( x ) - s _ { N } ( x ) \right| \leqslant 2 \varepsilon$ for all $x \in [ 0,1 ]$. Then bound the modulus of $s ( x ) - s ( 1 ) = \left( s ( x ) - s _ { N } ( x ) \right) + \left( s _ { N } ( x ) - s _ { N } ( 1 ) \right) + \left( s _ { N } ( 1 ) - s ( 1 ) \right)$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a complex sequence. We assume that the series $\sum a _ { n }$ converges. For $n \in \mathbb { N }$, we denote $r _ { n } = \sum _ { k = n + 1 } ^ { + \infty } a _ { k }$ and we define the functions $s _ { n }$ and $s$ from $[ 0,1 ]$ to $\mathbb { C }$ by $s _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }$ and $s ( x ) = \sum _ { k = 0 } ^ { + \infty } a _ { k } x ^ { k }$.
Application: recover the result from question II.B.3.

Let $\theta \in \mathbb { R }$. Determine the power series expansion of the function $$x \mapsto \frac { 1 - x ^ { 2 } } { x ^ { 2 } - 2 x \cos \theta + 1 }$$ on an interval to be specified.

Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b \neq 0$. We denote $d = R(a^2 + b)$. We call $W$ the sequence $W = ((a+d)^n)_{n \in \mathbb{N}}$ and $W'$ the sequence $W' = ((a-d)^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $d$, $W$ and $W'$.

Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b = 0$ and $a \neq 0$. We denote $W$ and $W'$ the sequences $W = (a^n)_{n \in \mathbb{N}}$ and $W' = (na^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $a$, $W$ and $W'$.

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Explicitly write $V_1(z)$, $V_2(z)$ and $V_3(z)$ and determine their roots in $\mathbb{C}$.

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Show that, for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, we have $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$ One may proceed by induction.

We use the notation $R$ introduced in part I and $V_n(z) = U_{n+1}(z,-1)$. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$, with $r$, $s$, $t$, $h$ as defined in II.C.1. What can be said about the radius of convergence of the power series $Z \mapsto \sum_{n=0}^{+\infty} V_n(z) Z^n$? We denote $g_z$ its sum.

We use the notation $V_n(z) = U_{n+1}(z,-1)$ and $g_z$ the sum of the power series $\sum_{n=0}^{+\infty} V_n(z) Z^n$. When it makes sense, calculate $\left(1 - 2zZ + Z^2\right) g_z(Z)$.