UFM Additional Further Pure

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grandes-ecoles 2021 Q18a Evaluation of a Finite or Infinite Sum View
Show that for all $x \in ]0,1[$: $$\frac{\pi}{\sin(\pi x)} = \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+x} + \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+1-x}.$$
grandes-ecoles 2021 Q22 Power Series Expansion and Radius of Convergence View
Let $f \in L^1(\mathbb{R})$, $\lambda \in \mathbb{R}_+^*$ and let $g$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that $g(x) = f(\lambda x)$ for all real $x$. Show that $g \in L^1(\mathbb{R})$ and, for all real $\xi$, express $\hat{g}(\xi)$ in terms of $\hat{f}$, $\xi$ and $\lambda$.
grandes-ecoles 2021 Q22 Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.
grandes-ecoles 2021 Q23 Evaluation of a Finite or Infinite Sum View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. The convolution product of $f$ and $g$ is defined by $$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(t) g(x-t) \,\mathrm{d}t$$
Show that $f * g$ is defined on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad (f * g)(x) = \int_{-\infty}^{+\infty} f(x-t) g(t) \,\mathrm{d}t = (g * f)(x)$$
grandes-ecoles 2021 Q23 Evaluation of a Finite or Infinite Sum View
Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is defined on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad (f*g)(x) = \int_{-\infty}^{+\infty} f(x-t)g(t)\,\mathrm{d}t = (g*f)(x)$$
grandes-ecoles 2021 Q23 Limit Evaluation Involving Sequences View
We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Deduce that $$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$
grandes-ecoles 2021 Q23 Limit Evaluation Involving Sequences View
Deduce that $$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$
grandes-ecoles 2021 Q23 Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.
grandes-ecoles 2021 Q24 Proof of Inequalities Involving Series or Sequence Terms View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f * g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.
grandes-ecoles 2021 Q24 Proof of Inequalities Involving Series or Sequence Terms View
Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f*g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.
grandes-ecoles 2021 Q24 Functional Equations and Identities via Series View
We classify cycles of length $k$ into three subsets:
  • the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
  • the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
  • the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then $$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$
grandes-ecoles 2021 Q24 Functional Equations and Identities via Series View
Cycles of length $k$ are classified into three subsets: the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once; the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice; the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then $$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$
grandes-ecoles 2021 Q24 Recurrence Relations and Sequence Properties View
In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, consider the function $Q_n : \left|\,\begin{array}{ccl} [-1,1] & \rightarrow & \mathbb{R} \\ x & \mapsto & \cos(n \arccos(x)) \end{array}\right.$.
Calculate $Q_0$, $Q_1$ and, for all $n \in \mathbb{N}$, express simply $Q_{n+2}$ in terms of $Q_{n+1}$ and $Q_n$.
grandes-ecoles 2021 Q25 Uniform or Pointwise Convergence of Function Series/Sequences View
We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f * g)^{(k)} = f * (g^{(k)})$.
grandes-ecoles 2021 Q25 Uniform or Pointwise Convergence of Function Series/Sequences View
Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f*g)^{(k)} = f * (g^{(k)})$.
grandes-ecoles 2021 Q25 Recurrence Relations and Sequence Properties View
In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, consider the function $Q_n(x) = \cos(n \arccos(x))$ on $[-1,1]$.
Deduce that, for all $n \in \mathbb{N}$, $Q_n$ is polynomial and determine its degree and leading coefficient.
grandes-ecoles 2021 Q26 Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, let $Q_n$ denote the polynomial of $\mathbb{R}[X]$ that coincides with $x \mapsto \cos(n \arccos(x))$ on $[-1,1]$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $$\begin{cases} p_0 = Q_0 \\ \forall n \in \mathbb{N}^*, \quad p_n = \dfrac{1}{2^{n-1}} Q_n \end{cases}$$
grandes-ecoles 2021 Q27 Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. The orthogonal polynomials associated with $w$ satisfy $p_n = \frac{1}{2^{n-1}} Q_n$ for $n \geqslant 1$, where $Q_n(x) = \cos(n \arccos(x))$.
For $n \in \mathbb{N}$, explicitly determine the points $(x_j)_{0 \leqslant j \leqslant n}$ of $I$ such that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ has maximal order.
grandes-ecoles 2021 Q28 Power Series Expansion and Radius of Convergence View
We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$ and with $\alpha_0 = 1$. We denote by $S$ the sum of this power series on its disk of convergence.
Show that there exists a real number $q > 0$ such that $\forall n \in \mathbb{N}, |\alpha_n| \leqslant q^n$.
grandes-ecoles 2021 Q29 Recurrence Relations and Sequence Properties View
Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$. For all $n \in \mathbb { N }$, show that $U _ { n }$ is monic of degree $n$, and determine the value of $U _ { n } ( 0 )$.
grandes-ecoles 2021 Q29 Properties and Manipulation of Power Series or Formal Series View
We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$ and with $\alpha_0 = 1$, and sum $S$. We assume that $\frac{1}{S}$ is expandable as a power series in a neighbourhood of 0 and we denote by $\sum_{n \geqslant 0} \beta_n z^n$ its expansion.
Calculate $\beta_0$ and, for all $n \in \mathbb{N}^*$, express $\beta_n$ in terms of $\alpha_1, \ldots, \alpha_n, \beta_1, \ldots, \beta_{n-1}$. Deduce that $$\forall n \in \mathbb{N}, \quad |\beta_n| \leqslant (2q)^n.$$
grandes-ecoles 2021 Q30 Evaluation of a Finite or Infinite Sum View
Deduce that, for every polynomial $P \in \mathbb{R}[X]$, $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} P\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} P(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$
grandes-ecoles 2021 Q30 Evaluation of a Finite or Infinite Sum View
Deduce that, for any polynomial $P \in \mathbb{R}[X]$, $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} P\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} P(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$
grandes-ecoles 2021 Q30 Recurrence Relations and Sequence Properties View
Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$. Let $\theta \in \mathbb { R }$. Show that $\forall n \in \mathbb { N } , U _ { n } \left( 4 \cos ^ { 2 } \theta \right) \sin \theta = \sin ( ( 2 n + 1 ) \theta )$.
grandes-ecoles 2021 Q30 Properties and Manipulation of Power Series or Formal Series View
We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$, $\alpha_0 = 1$, and sum $S$.
Show that $\frac{1}{S}$ is expandable as a power series in a neighbourhood of 0.

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