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

2020 centrale-maths2__psi

39 maths questions

Q1 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Justify that the mapping $f$ establishes a bijection from the interval $\left[ - 1 , + \infty \left[ \right. \right.$ onto the interval $\left[ - e ^ { - 1 } , + \infty [ \right.$, where $f : \mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto x\mathrm{e}^{x}$.

Q2 Chain Rule Derivative of Inverse Functions View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. That is, for every real $x \geqslant -\mathrm{e}^{-1}$, $W(x)$ is the unique solution of $f(t) = x$ with $t \in [-1,+\infty[$. Justify that $W$ is continuous on $\left[ - \mathrm { e } ^ { - 1 } , + \infty \left[ \right. \right.$ and is of class $\mathcal { C } ^ { \infty }$ on $] - \mathrm { e } ^ { - 1 } , + \infty [$.

Q3 Chain Rule Derivative of Inverse Functions View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Explicitly determine $W ( 0 )$ and $W ^ { \prime } ( 0 )$.

Q4 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Determine an equivalent of $W ( x )$ as $x \rightarrow 0$ as well as an equivalent of $W ( x )$ as $x \rightarrow + \infty$.

Q5 Curve Sketching Sketching a Curve from Analytical Properties View

Let $f(x) = xe^x$ and let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$. Sketch, on the same diagram, the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { W }$ representing the functions $f$ and $W$. Specify the tangent lines to the two curves at the point with abscissa 0 as well as the tangent line to $\mathcal { C } _ { W }$ at the point with abscissa $- \mathrm { e } ^ { - 1 }$.

Q6 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $\left. ]0,1 \right]$?

Q7 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Q8 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Let $f(x) = xe^x$. Prove that the mapping $f$ establishes a bijection from the interval $] - \infty , - 1 ]$ onto the interval $\left[ - \mathrm { e } ^ { - 1 } , 0 [ \right.$. In the rest of the problem, the inverse of this bijection is denoted $V$.

Q9 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the equation with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } = m \tag{I.1}$$
Determine, as a function of $m$, the number of solutions of (I.1). Explicitly express the possible solutions using the functions $V$ and $W$.

Q10 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the inequality with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } \leqslant m \tag{I.2}$$
Using the functions $V$ and $W$, determine, according to the values of $m$, the solutions of (I.2). Illustrate graphically the different cases.

Q11 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively, where $f(x) = xe^x$. For non-zero real parameters $a$ and $b$, we consider the equation with unknown $x \in \mathbb { R }$
$$\mathrm { e } ^ { a x } + b x = 0 \tag{I.3}$$
Determine, according to the values of $a$ and $b$, the number of solutions of (I.3). Explicitly express the possible solutions using the functions $V$ and $W$.

Q12 Poisson distribution View

Q13 Poisson distribution View

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ Using Markov's inequality, prove that if $p \leqslant 2 \frac { 1 - \alpha } { \lambda }$, then condition (II.1) is satisfied.

Q14 Poisson distribution View

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ We set $x = -(\lambda p + 1)$. Prove that condition (II.1) is equivalent to the condition $$x \mathrm { e } ^ { x } \leqslant - \alpha \mathrm { e } ^ { - 1 }.$$

Q15 Poisson distribution View

Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, discuss according to the position of $\lambda$ with respect to $-1 - V\left(-\alpha \mathrm{e}^{-1}\right)$ the existence of a largest real $p \in ]0,1[$ satisfying condition (II.1).

Q16 Binomial Distribution Compute Expectation, Variance, or Standard Deviation View

Q17 Modelling and Hypothesis Testing View

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits, and $X$ follows the distribution determined in Q16. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ Using Markov's inequality, prove that if $r \leqslant 2 \frac{1-\alpha}{1-p}$, then condition (II.2) is satisfied.

Q18 Discrete Probability Distributions Binomial Distribution Identification and Application View

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ We set $a = \frac{p \ln(p)}{p-1}$ and $x = r\ln(p) - a$. Prove that condition (II.2) is equivalent to the condition $$x \mathrm{e}^{x} \leqslant -\alpha a \mathrm{e}^{-a}.$$

Q19 Discrete Probability Distributions Binomial Distribution Identification and Application View

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$ and $x = r\ln(p) - a$, condition (II.2) is equivalent to $xe^x \leqslant -\alpha a e^{-a}$. Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, study the existence of a largest natural integer $r$ satisfying condition (II.2).

Q20 Binomial Distribution Find n or Threshold from Cumulative Probability Condition View

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$. Let $V$ and $W$ be the Lambert functions defined in Part I. When it exists, express the largest natural integer $r$ satisfying condition (II.2) as a function of $p$, $\alpha$ and $a$ using one of the functions $V$ or $W$.

Q21 Roots of polynomials Polynomial evaluation, interpolation, and remainder View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ We denote by $\mathbb{C}_n[X]$ the $\mathbb{C}$-vector space of polynomials with complex coefficients and degree at most $n$. Prove that the family $(A_0, \ldots, A_n)$ is a basis of $\mathbb{C}_n[X]$.

Q22 Roots of polynomials Multiplicity and derivative analysis of roots View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Prove that for all $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

Q23 Sequences and Series Recurrence Relations and Sequence Properties View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Using the result $A_k'(X) = A_{k-1}(X-a)$, deduce, for $j$ and $k$ elements of $\llbracket 0, n \rrbracket$, the value of $A_k^{(j)}(ja)$. Distinguish according to whether $j < k$, $j = k$ or $j > k$.

Q24 Sequences and Series Functional Equations and Identities via Series View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Let $P$ be an element of $\mathbb{C}_n[X]$ and let $\alpha_0, \ldots, \alpha_n$ be complex numbers such that $$P = \sum_{k=0}^{n} \alpha_k A_k.$$ Prove that, for all $j \in \llbracket 0, n \rrbracket$, $\alpha_j = P^{(j)}(ja)$.

Q25 Sequences and Series Functional Equations and Identities via Series View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Using the result of Question 24, deduce Abel's binomial identity: $$\forall (a, x, y) \in \mathbb{C}^3, \quad (x+y)^n = y^n + \sum_{k=1}^{n} \binom{n}{k} x(x - ka)^{k-1}(y + ka)^{n-k}.$$

Q26 Sequences and Series Functional Equations and Identities via Series View

We consider a natural integer $n$ and a complex number $a$. Using Abel's binomial identity $$\forall (a, x, y) \in \mathbb{C}^3, \quad (x+y)^n = y^n + \sum_{k=1}^{n} \binom{n}{k} x(x - ka)^{k-1}(y + ka)^{n-k},$$ establish the relation $$\forall (a, y) \in \mathbb{C}^2, \quad ny^{n-1} = \sum_{k=1}^{n} \binom{n}{k} (-ka)^{k-1}(y + ka)^{n-k}.$$

Q27 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Q28 Taylor series Extract derivative values from a given series View

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Justify that the function $S$ is of class $\mathcal{C}^\infty$ on $]-R, R[$ and, for every integer $n \in \mathbb{N}$, express $S^{(n)}(0)$ as a function of $n$.

Q29 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Q30 Sequences and series, recurrence and convergence Series convergence and power series analysis View

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Prove that $$\forall x \in ]-R, R[, \quad x(1 + S(x))S'(x) = S(x).$$ One may use the result from Question 26.

Q31 Differential equations Verification that a Function Satisfies a DE View

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. We consider the function $$h : \begin{array}{ccc} ]-R,R[ & \rightarrow & \mathbb{R} \\ x & \mapsto & S(x)\mathrm{e}^{S(x)} \end{array}$$ Prove that $h$ is a solution on $]-R, R[$ of the differential equation $xy' - y = 0$.

Q32 Differential equations First-Order Linear DE: General Solution View

Solve the differential equation $xy' - y = 0$ on each of the intervals $]0, R[$ and $]-R, 0[$ then on the interval $]-R, R[$.

Q33 Sequences and series, recurrence and convergence Convergence proof and limit determination View

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Let $W$ be the Lambert function defined in Part I (inverse of $f|_{[-1,+\infty[}$ where $f(x)=xe^x$). Using the results of Questions 31 and 32, deduce that $$\forall x \in ]-R, R[, \quad S(x) = W(x).$$

Q34 Sequences and Series Power Series Expansion and Radius of Convergence View

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. It has been shown that $S(x) = W(x)$ for all $x \in ]-R,R[$. Does this result remain true on $[-R, R]$?

Q35 Differentiating Transcendental Functions Existence and uniqueness of solutions involving transcendental equations View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Prove that, for every positive real $x$, $W(x)$ is a fixed point of $\phi_x$, that is, a solution of the equation $\phi_x(t) = t$.

Q36 Differentiating Transcendental Functions Compute derivative of transcendental function View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ Prove that, for every positive real $x$, the function $\phi_x$ is of class $\mathcal{C}^2$ on $\mathbb{R}$ and that $$\forall t \in \mathbb{R}, \quad 0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}.$$

Q37 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Using the fact that $0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}$ for all $t \in \mathbb{R}$, deduce that $$\forall x \in [0, \mathrm{e}], \quad \forall n \in \mathbb{N}, \quad |w_n(x) - W(x)| \leqslant \left(\frac{x}{\mathrm{e}}\right)^n |1 - W(x)|.$$

Q38 Sequences and series, recurrence and convergence Sequence of functions convergence View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. For every real $a \in ]0, \mathrm{e}[$, justify that the sequence of functions $(w_n)$ converges uniformly on $[0, a]$ to the function $W$.

Q39 Sequences and series, recurrence and convergence Sequence of functions convergence View

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?