Given that $f ( x )$ is an odd function with domain $( - \infty , + \infty )$ satisfying $f ( 1 - x ) = f ( 1 + x )$. If $f ( 1 ) = 2$, then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 30 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$

Let $f ( x )$ be an odd function with domain $( - \infty , + \infty )$, satisfying $f ( 1 - x ) = f ( 1 + x )$ and $f ( 1 ) = 2$. Then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 50 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$

Executing the flowchart on the right, if the input $\varepsilon$ is 0.01, then the output value of $s$ equals
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$

After completing its lunar exploration mission, the Chang'e-2 satellite continued deep space exploration and became China's first artificial planet orbiting the sun. To study the ratio of Chang'e-2's orbital period around the sun to Earth's orbital period around the sun, the sequence $\{b_n\}$ is used: $b_1 = 1 + \frac{1}{a_1}, b_2 = 1 + \frac{1}{a_1 + \frac{1}{a_2}}, b_3 = 1 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}, \cdots$, and so on, where $a_k \in \mathbf{N}^* (k = 1,2,\cdots)$. Then
A. $b_1 < b_5$
B. $b_3 < b_8$
C. $b_6 < b_2$
D. $b_4 < b_7$

Given that $f(x), g(x)$ have domain $\mathbf{R}$, and $f(x) + g(2-x) = 5, g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, and $g(2) = 4$, then $\sum_{k=1}^{22} f(k) =$
A. $-21$
B. $-22$
C. $-23$
D. $-24$

Given $M = \left\{ k \mid a _ { k } = b _ { k } \right\}$, where $a _ { n }, b _ { n }$ are not constant sequences and all terms are distinct. Which of the following is correct? \_\_\_
(1) If $a _ { n }, b _ { n }$ are both arithmetic sequences, then $M$ has at most one element;
(2) If $a _ { n }, b _ { n }$ are both geometric sequences, then $M$ has at most three elements;
(3) If $a _ { n }$ is an arithmetic sequence and $b _ { n }$ is a geometric sequence, then $M$ has at most three elements;
(4) If $a _ { n }$ is monotonically increasing and $b _ { n }$ is monotonically decreasing, then $M$ has at most one element.

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that $K$ is a compact subset of $\mathbf{R}^3$ for its usual topology.
b) Prove that $K$ is convex, that is, for every real $t\in[0,1]$ and every pair $(u,v)$ of elements of $K$, $tu+(1-t)v$ belongs to $K$.
c) Establish that, if $(a,b,c)\in\mathbf{C}^3$, $\widehat{abc}$ is a compact convex subset of $\mathbf{C}$ equipped with its usual topology.
d) With the same notation prove the existence of: $$\delta(\widehat{abc}) = \max\left\{|z'-z| / (z,z')\in\widehat{abc}^2\right\}$$

We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that, if we fix $z\in\mathbf{C}$ and $(a,b,c)\in\mathbf{C}^3$: $$\max\left\{|z'-z| / z'\in\widehat{abc}\right\} = \max(|z-a|,|z-b|,|z-c|)$$
b) Deduce from this a simple expression for $\delta(\widehat{abc})$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:

• [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
• [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.

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 $j \in \mathbb{N}$.
Show that the numerical series $\sum_{n \in \mathbb{N}} n^j \alpha_n$ is convergent.

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 $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.

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: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
Show that $f$ is of class $C^\infty$ on $[-1,1]$.

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 $b_n(\widetilde{f})$?

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
Let $f \in C^\infty([-1,1])$.
Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Let $f \in C^\infty([-1,1])$.
Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that $$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$ for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.

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.

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 $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. For a function $h \in C([-1,1])$, $\widetilde{h}$ denotes the $2\pi$-periodic function $\theta \mapsto h(\cos(\theta))$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
The purpose of this question is to show that the function $f$ is of class $C^\infty$.
a) Let $k \in \mathbb{N}^*$. Show that, for $P \in E_{k-1}$, $a_k(\widetilde{f}) = a_k(\widetilde{f-P})$.
b) Deduce that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ of Fourier coefficients of the function $\widetilde{f}$ has rapid decay.
c) Conclude.

For every element $x$ of $E$, we denote by $h(x)$ the application from $E$ to $E$ such that $\forall y \in E, h(x)(y) = \varphi(x,y)$.
a) Show that, for all $x$ in $E$, $h(x)$ is an element of the dual of $E$, denoted $E^{*}$.
b) Show that $h$ is a linear application from $E$ to $E^{*}$.

Deduce the nature of the Riemann series $\sum _ { n \geqslant 1 } \frac { 1 } { n ^ { \alpha } }$ according to the value of $\alpha \in \mathbb { R }$.

For every real $\alpha > 1$, show that $1 \leqslant S ( \alpha ) \leqslant 1 + \frac { 1 } { \alpha - 1 }$, where $S ( \alpha ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { \alpha } }$.

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 $$R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + \frac { 1 } { 2 n ^ { \alpha } } + O \left( \frac { 1 } { n ^ { \alpha + 1 } } \right)$$