grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2021 centrale-maths2__pc

38 maths questions

Q1 Numerical integration Quadrature Formula Construction and Order Determination View
In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the order of the quadrature formula $I_0(f) = f(0)$ and represent graphically the associated error $e(f)$.
Q2 Numerical integration Quadrature Formula Construction and Order Determination View
In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the order of the quadrature formula $I_0(f) = f(1/2)$ and represent graphically the associated error $e(f)$.
Q3 Numerical integration Quadrature Formula Construction and Order Determination View
In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the coefficients $\lambda_0, \lambda_1, \lambda_2$ so that the formula $I_2(f) = \lambda_0 f(0) + \lambda_1 f(1/2) + \lambda_2 f(1)$ is exact on $\mathbb{R}_2[X]$. Is this quadrature formula of order 2?
Q4 Matrices Linear Transformation and Endomorphism Properties View
Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and a continuous function $f$ from $I$ to $\mathbb{R}$.
Show that the linear map $\varphi : \left|\,\begin{array}{ccl} \mathbb{R}_n[X] & \rightarrow & \mathbb{R}^{n+1} \\ P & \mapsto & \left(P(x_0), P(x_1), \ldots, P(x_n)\right) \end{array}\right.$ is an isomorphism.
Q5 Proof Existence Proof View
Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$.
Show that, for all $i \in \llbracket 0, n \rrbracket$, there exists a unique polynomial $L_i \in \mathbb{R}_n[X]$ such that $$\forall j \in \llbracket 0, n \rrbracket, \quad L_i(x_j) = \begin{cases} 0 & \text{if } j \neq i, \\ 1 & \text{if } j = i. \end{cases}$$
Q6 Proof Proof of Set Membership, Containment, or Structural Property View
Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and the polynomials $L_0, \ldots, L_n$ defined in Q5.
Show that $(L_0, \ldots, L_n)$ is a basis of $\mathbb{R}_n[X]$.
Q7 Numerical integration Lagrange Basis Recovery of Quadrature Weights View
Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and the Lagrange basis $(L_0, \ldots, L_n)$ associated with these points.
Suppose that, for all $k \in \mathbb{N}$, the map $x \mapsto x^k w(x)$ is integrable on $I$. Show that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ is exact on $\mathbb{R}_n[X]$ if and only if $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x.$$
Q8 Numerical integration Lagrange Basis Recovery of Quadrature Weights View
We consider the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$. Determine the Lagrange basis associated with the points $(0, 1/2, 1)$ and thus recover the coefficients of the quadrature formula $I_2(f)$ from question 3.
Q9 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We assume that $I = [a,b]$ with $a < b$, $\forall x \in I, w(x) = 1$ (general weight $w$ in the formula for $e(f)$), and that $f$ is of class $\mathcal{C}^{m+1}$ on $I$, where $m$ is the order of the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$.
For every natural number $m$, consider the function $\varphi_m : \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $$\forall (x,t) \in \mathbb{R}^2, \quad \varphi_m(x,t) = \begin{cases} (x-t)^m & \text{if } x \geqslant t, \\ 0 & \text{if } x < t. \end{cases}$$
Using the Taylor formula with integral remainder, show that $e(f) = e(R_m)$, where $R_m$ is defined by $$\forall x \in [a,b], \quad R_m(x) = \frac{1}{m!} \int_a^b \varphi_m(x,t) f^{(m+1)}(t)\,\mathrm{d}t.$$
Q10 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We assume that $I = [a,b]$ with $a < b$, and that $f$ is of class $\mathcal{C}^{m+1}$ on $I$, where $m \geqslant 1$ is the order of the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$.
For every natural number $m$, consider the function $\varphi_m : \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $$\forall (x,t) \in \mathbb{R}^2, \quad \varphi_m(x,t) = \begin{cases} (x-t)^m & \text{if } x \geqslant t, \\ 0 & \text{if } x < t. \end{cases}$$
Deduce that, if $m \geqslant 1$, $$e(f) = \frac{1}{m!} \int_a^b K_m(t) f^{(m+1)}(t)\,\mathrm{d}t$$ where the function $K_m : [a,b] \rightarrow \mathbb{R}$ is defined by $$\forall t \in [a,b], \quad K_m(t) = e\left(x \mapsto \varphi_m(x,t)\right) = \int_a^b \varphi_m(x,t) w(x)\,\mathrm{d}x - \sum_{j=0}^n \lambda_j \varphi_m(x_j, t).$$ You may use the following admitted result: for every continuous function $g : [a,b]^2 \rightarrow \mathbb{R}$, we have $$\int_a^b \left(\int_a^b g(x,t)\,\mathrm{d}t\right)\mathrm{d}x = \int_a^b \left(\int_a^b g(x,t)\,\mathrm{d}x\right)\mathrm{d}t.$$
Q11 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We assume that $I = [0,1]$, $\forall x \in I, w(x) = 1$, and we consider the quadrature formula $$I_1(g) = \frac{g(0) + g(1)}{2},$$ which is of order $m = 1$.
Calculate the associated Peano kernel $t \mapsto K_1(t)$ and show that, for every function $g$ of class $\mathcal{C}^2$ from $[0,1]$ to $\mathbb{R}$, we have the following bound on the associated quadrature error: $$|e(g)| \leqslant \frac{1}{12} \sup_{x \in [0,1]} |g''(x)|.$$
Q12 Numerical integration Composite Rule Error Decomposition View
We consider the case of an arbitrary segment $I = [a,b]$ (with $a < b$), subdivided into $n+1$ equidistant points $a_0, \ldots, a_n$: $$\forall i \in \llbracket 0, n \rrbracket, \quad a_i = a + ih,$$ where $h = \frac{b-a}{n}$ is the step of the subdivision. The trapezoidal rule is $$T_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2}.$$
Represent graphically $T_n(f)$.
Q13 Numerical integration Composite Rule Error Decomposition View
We consider the trapezoidal rule on $I = [a,b]$: $$T_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2},$$ where $a_i = a + ih$ and $h = \frac{b-a}{n}$, with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$.
Suppose that $f$ is a function of class $\mathcal{C}^2$ from $[a,b]$ to $\mathbb{R}$. Show that $$e_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} e(g_i)$$ where $e$ is the error associated with the quadrature formula $I_1$ studied in question 11 and the $g_i : [0,1] \rightarrow \mathbb{R}$ are functions to be specified.
Q14 Numerical integration Quadrature Error Bound Derivation View
We consider the trapezoidal rule on $I = [a,b]$ with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$, where $f$ is of class $\mathcal{C}^2$.
Deduce the error bound $$\left|e_n(f)\right| \leqslant \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} |f''(x)|.$$
Q15 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that, for all functions $f$ and $g$ in $E$, the product $fgw$ is integrable on $I$. You may use the inequality $\forall (a,b) \in \mathbb{R}^2, |ab| \leqslant \frac{1}{2}(a^2 + b^2)$, after justifying it.
Q16 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that $E$ is an $\mathbb{R}$-vector space.
Q17 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$. For all functions $f$ and $g$ in $E$, we set $$\langle f, g \rangle = \int_I f(x) g(x) w(x)\,\mathrm{d}x.$$
Show that we thus define an inner product on $E$.
Q18 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We assume that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, and $\langle p_i, p_j \rangle = 0$ for $i \neq j$).
Let $n \in \mathbb{N}^*$. We denote by $x_1, \ldots, x_k$ the distinct roots of $p_n$ that are in $\mathring{I}$ and $m_1, \ldots, m_k$ their respective multiplicities. We consider the polynomial $$q(X) = \prod_{i=1}^k (X - x_i)^{\varepsilon_i}, \quad \text{with } \varepsilon_i = \begin{cases} 1 & \text{if } m_i \text{ is odd} \\ 0 & \text{if } m_i \text{ is even.} \end{cases}$$
By studying $\langle p_n, q \rangle$, show that $p_n$ has $n$ distinct roots in $\mathring{I}$.
Q19 Proof Direct Proof of an Inequality View
Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where $n \in \mathbb{N}$, $\lambda_0, \ldots, \lambda_n \in \mathbb{R}$ and $x_0 < x_1 < \cdots < x_n$ are $n+1$ distinct points in $I$. We assume that the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Thus, the formula $I_n(f)$ is of order $m \geqslant n$.
By reasoning with the polynomial $\prod_{i=0}^n (X - x_i)$, show that $m \leqslant 2n+1$.
Q20 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $m = 2n+1$ if and only if the $x_i$ are the roots of $p_{n+1}$.
Q21 Proof Computation of a Limit, Value, or Explicit Formula View
We consider the case where $I = [-1,1]$ and $w(x) = 1$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, orthogonal for $\langle f, g \rangle = \int_{-1}^1 f(x)g(x)\,\mathrm{d}x$).
Determine the first four orthogonal polynomials $(p_0, p_1, p_2, p_3)$ associated with the weight $w$.
Q23 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$.
Show that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$.
Q24 Sequences and Series 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 $Q_n$.
Q25 Sequences and Series 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.
Q26 Sequences and Series 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 $$\begin{cases} p_0 = Q_0 \\ \forall n \in \mathbb{N}^*, \quad p_n = \dfrac{1}{2^{n-1}} Q_n \end{cases}$$
Q27 Sequences and Series 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.
Q28 Sequences and Series 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 $\forall n \in \mathbb{N}, |\alpha_n| \leqslant q^n$.
Q29 Sequences and Series 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 $$\forall n \in \mathbb{N}, \quad |\beta_n| \leqslant (2q)^n.$$
Q30 Sequences and Series 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.
Q31 Sequences and Series Power Series Expansion and Radius of Convergence View
Using the results of questions 28--30, show that there exists a unique complex sequence $(b_n)_{n \in \mathbb{N}}$ and a real number $r > 0$ such that, for all $z \in \mathbb{C}$, $$0 < |z| < r \Rightarrow \frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n.$$
Q32 Sequences and Series Functional Equations and Identities via Series View
Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ (valid for $0 < |z| < r$), by performing a Cauchy product, show that $b_0 = 1$ and, for all integer $n \geqslant 2$, $$\sum_{p=0}^{n-1} \binom{n}{p} b_p = 0.$$
Q33 Sequences and Series Evaluation of a Finite or Infinite Sum View
Using the relation $b_0 = 1$ and $\sum_{p=0}^{n-1} \binom{n}{p} b_p = 0$ for all integer $n \geqslant 2$, deduce the value of $b_1, b_2, b_3$ and $b_4$.
Q34 Sequences and Series Functional Equations and Identities via Series View
Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ and a parity argument, show that $b_{2p+1} = 0$ for all integer $p \geqslant 1$.
Q35 Sequences and Series Evaluation of a Finite or Infinite Sum View
The polynomials $B_m$ are defined by $$\forall m \in \mathbb{N}, \quad B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}.$$
Determine $B_0, B_1, B_2$ and $B_3$.
Q36 Sequences and Series Functional Equations and Identities via Series View
The polynomials $B_m$ are defined by $$\forall m \in \mathbb{N}, \quad B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}.$$
Show that, for all integer $m \geqslant 2$, $B_m(1) = b_m$, then that, for all integer $m \geqslant 1$, $B_m' = m B_{m-1}$.
Q37 Sequences and Series Functional Equations and Identities via Series View
We fix an integer $n \in \mathbb{N}^*$ and consider a function $g : [0,n] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$. The polynomial $B_1$ is as defined in the sequence $(B_m)$.
Show that $$\int_0^n g(x)\,\mathrm{d}x = \sum_{k=0}^{n-1} \frac{g(k) + g(k+1)}{2} - \int_0^n B_1(x - \lfloor x \rfloor) g'(x)\,\mathrm{d}x.$$
Q38 Sequences and Series Functional Equations and Identities via Series View
We fix an integer $n \in \mathbb{N}^*$ and consider a function $g : [0,n] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$. The polynomials $B_m$ are defined by $B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}$.
Deduce that for all integer $m \geqslant 2$, $$\int_0^n g(x)\,\mathrm{d}x = \sum_{k=0}^{n-1} \frac{g(k)+g(k+1)}{2} + \sum_{p=2}^m \frac{(-1)^{p-1} b_p}{p!}\left(g^{(p-1)}(n) - g^{(p-1)}(0)\right) + \frac{(-1)^m}{m!} \int_0^n B_m(x - \lfloor x \rfloor) g^{(m)}(x)\,\mathrm{d}x.$$
Q39 Numerical integration Euler–Maclaurin / Asymptotic Expansion of Trapezoidal Rule View
We consider a function $f : [a,b] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$ and the trapezoidal method $$T_n(f) = h \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2},$$ where $h = \frac{b-a}{n}$ and $\forall i \in \llbracket 0, n-1 \rrbracket, a_i = a + ih$.
Using the result of question 38, show that, for all integer $m \geqslant 1$, $$\int_a^b f(x)\,\mathrm{d}x = T_n(f) - \sum_{p=1}^m \frac{\gamma_{2p}}{n^{2p}} + \rho_{2m}(n)$$ where the coefficients $\gamma_{2p}$ are given by $$\gamma_{2p} = \frac{(b-a)^{2p} b_{2p}}{(2p)!}\left(f^{(2p-1)}(b) - f^{(2p-1)}(a)\right)$$ and $\rho_{2m}(n)$ is a remainder integral satisfying the bound $$|\rho_{2m}(n)| \leqslant \frac{C_{2m}}{n^{2m}}$$ where $C_{2m}$ is a constant to be determined depending only on $m$, $a$ and $b$.