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
2015 x-ens-maths2__mp

31 maths questions

Q1a Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View
Show that $\Gamma$ is well defined and that for all $y > 0 , y \Gamma ( y ) = \Gamma ( y + 1 )$. Deduce that, for all $n \in \mathbb { N } , \Gamma ( n + 1 ) = n !$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$ and that $\Gamma \left( \frac { 1 } { 2 } \right) = \sqrt { \pi }$.
Q1b Reduction Formulae Perform a Change of Variable or Transformation on a Parametric Integral View
Show that for all $y > 0$, we have $\Gamma ( y ) = y ^ { - 1 } \int _ { 0 } ^ { + \infty } e ^ { - t } t ^ { y } d t$, then that
$$\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { - 1 } ^ { + \infty } e ^ { - y \phi ( s ) } d s$$
where $\phi$ is the function defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$.
Q2a Reduction Formulae Bound or Estimate a Parametric Integral View
We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\delta > 0$ and $\alpha \in \mathbb { R }$. Show that for all $x > 0$, the function $t \mapsto e ^ { - t / x } t ^ { \alpha }$ is integrable on $[ \delta , + \infty [$ and that for all $n \in \mathbb { N }$, we have: $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } t ^ { \alpha } d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$ Deduce that for all $n \in \mathbb { N }$, $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } \rho _ { N } ( t ) d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$
Q2b Reduction Formulae Bound or Estimate a Parametric Integral View
We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\varepsilon > 0$. Show the existence of $\delta > 0$ and a constant $C ^ { \prime }$ independent of $\varepsilon$ and $\delta$ such that $$\forall x > 0 , \quad \left| \int _ { 0 } ^ { \delta } e ^ { - t / x } \rho _ { N } ( t ) d t \right| \leqslant C ^ { \prime } \varepsilon x ^ { ( N + \lambda ) / \mu }$$
Q2c Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View
We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
Deduce that $$\int _ { 0 } ^ { + \infty } e ^ { - t / x } \rho _ { N } ( t ) d t = o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$
Q2d Reduction Formulae Establish an Integral Identity or Representation View
We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$
We denote $F$ the function defined by: $$F ( x ) = \int _ { 0 } ^ { + \infty } e ^ { - t / x } f ( t ) d t$$
Show that $F$ is well defined on $] 0 , + \infty [$ and that it satisfies the following asymptotic formula: $$F ( x ) = \sum _ { k = 0 } ^ { N } a _ { k } \Gamma \left( \frac { k + \lambda } { \mu } \right) x ^ { ( k + \lambda ) / \mu } + o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + } .$$
Q3a Curve Sketching Sketching a Curve from Analytical Properties View
We recall that the function $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Sketch the graph of $\phi$. Show that $\phi$ defines by restriction to the intervals $] - 1,0 [$ and $] 0 , + \infty [$ respectively
  • a bijection $\left. \phi _ { - } : \right] - 1,0 [ \rightarrow ] 0 , + \infty [$,
  • a bijection $\left. \phi _ { + } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$.
We denote $\left. \phi _ { - } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] - 1,0 \left[ \right.$ and $\left. \phi _ { + } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$ the inverse bijections.
Q3b Sequences and Series Power Series Expansion and Radius of Convergence View
We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Show that if $s \in ] - 1,1 [$, $$\phi ( s ) = s ^ { 2 } \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { k + 2 } s ^ { k }$$
We admit the existence of two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$, in the variable $q$, and of strictly positive radius of convergence, where $b _ { 1 } > 0$ and $c _ { 1 } < 0$, and such that we have, for $q$ in a neighborhood of 0 in $[ 0 , + \infty [$, $$\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 } .$$
Q3c Taylor series Taylor's formula with integral remainder or asymptotic expansion View
We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$, and that two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$ with strictly positive radius of convergence, $b_1 > 0$, $c_1 < 0$, satisfy $\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 }$ for $q$ near 0 in $[0,+\infty[$.
Calculate $b _ { 1 } , b _ { 2 } , b _ { 3 }$ and $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$. Deduce the following asymptotic expansions when $q \rightarrow 0 , q > 0$, for the functions $\phi _ { - } ^ { - 1 }$ and $\phi _ { + } ^ { - 1 }$ as well as their derivatives: $$\begin{array} { l l } \phi _ { + } ^ { - 1 } ( q ) = \sqrt { 2 q } + \frac { 2 q } { 3 } + \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) , & \phi _ { - } ^ { - 1 } ( q ) = - \sqrt { 2 q } + \frac { 2 q } { 3 } - \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) \\ \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) = \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } + \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) , & \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) = - \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } - \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) \end{array}$$
Q3d Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
We recall that $\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { - 1 } ^ { + \infty } e ^ { - y \phi ( s ) } d s$ where $\phi ( s ) = s - \ln(1+s)$, and that $\phi_-^{-1} : ]0,+\infty[ \to ]-1,0[$ and $\phi_+^{-1} : ]0,+\infty[ \to ]0,+\infty[$ are the inverse bijections of the restrictions of $\phi$.
Show that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { 0 } ^ { \infty } e ^ { - y q } \left( \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) - \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) \right) d q$$
Q3e Taylor series Taylor's formula with integral remainder or asymptotic expansion View
Using the results of the previous questions, deduce that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \left( \frac { 2 \pi } { y } \right) ^ { 1 / 2 } \left( 1 + \frac { 1 } { 12 y } + o \left( \frac { 1 } { y } \right) \right) \quad \text { when } y \rightarrow + \infty .$$
Q4 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $$F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$$
Show that $F$ is well defined and of class $\mathscr { C } ^ { \infty }$ on $] 0 , + \infty [$.
Q5 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\ S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$
Show that, for all $N \geqslant 1$ and all $x > 0 , F ( x ) = S _ { N } ( x ) + R _ { N } ( x )$.
Q6a Sequences and Series Convergence/Divergence Determination of Numerical Series View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Specify the domain of convergence of the series $\sum _ { k \geqslant 1 } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k }$ and show that the sequence $\left( R _ { N } ( x ) \right) _ { N \geqslant 1 }$ is not bounded.
Q6b Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that, if $N \in \mathbb { N } ^ { * }$ and $x > 0$, $$\left| R _ { N } ( x ) \right| \leqslant \left| r _ { N } ( x ) \right|$$ Deduce that $R _ { N + 1 } ( x ) = o \left( r _ { N } ( x ) \right)$ when $x \rightarrow 0$.
Q6c Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that the remainder is of the order of the first neglected term, that is, for all $N \geqslant 1$, $$R _ { N } ( x ) \sim r _ { N } ( x ) \quad \text { when } \quad x \rightarrow 0$$
Q6d Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }$$
Show that, for $0 < x < 1 / 2$, the sequence $\left( \left| r _ { N } ( x ) \right| \right) _ { N \geqslant 1 }$ is decreasing up to a certain rank, then increasing.
Q7a Sequences and Series Estimation or Bounding of a Sum View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\ S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Show that, if $N$ is even: $N = 2 M$ with $M \geqslant 1$, and if $0 < x \leqslant 1 / N$, we have $S _ { N } ( x ) \geqslant 0$ and $$E _ { N } ( x ) \leqslant \frac { N ! x ^ { N + 1 } } { \sum _ { \ell = 0 } ^ { M - 1 } ( 1 - ( 2 \ell + 1 ) x ) ( 2 \ell ) ! x ^ { 2 \ell + 1 } }$$
Q7b Sequences and Series Estimation or Bounding of a Sum View
We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$ The relative error is $E _ { N } ( x ) = \left| \frac { R _ { N } ( x ) } { F ( x ) } \right|$.
Verify that $E _ { 4 } \left( \frac { 1 } { 10 } \right) \leqslant 3.10 ^ { - 3 }$.
Q8 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We consider the space $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { d } \right)$ of functions $f : \mathbb { R } ^ { d } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of their variables, equipped with the uniform norm $\| f \| _ { \infty } = \sup _ { \theta \in \mathbb { R } ^ { d } } | f ( \theta ) |$. A trigonometric polynomial (in $d$ variables) is any function of the form $\theta \mapsto \sum _ { k \in K } c _ { k } e ^ { 2 i \pi k \cdot \theta }$ where $K$ is a finite subset of $\mathbb { Z } ^ { d }$. We work in dimension $d = 2$. The subspace $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ of $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ consists of functions of the form $\theta = \left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \sum _ { i = 1 } ^ { n } f _ { i } \left( \theta _ { 1 } \right) g _ { i } \left( \theta _ { 2 } \right)$, where $n \in \mathbb { N } ^ { * }$ and $f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$. We admit that trigonometric polynomials in one variable are dense in $\mathscr{C}_{\text{per}}(\mathbb{R})$.
Show that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$.
Q9 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $$\forall t \in ] - 1 / 2,1 / 2 ] , \quad \psi _ { j } ( t ) = \max ( 0,1 - j | t | ) .$$ For integers $0 \leqslant k < j$, the functions $\psi _ { j , k } : \mathbb { R } \rightarrow \mathbb { R }$ are defined by $$\forall t \in \mathbb { R } , \quad \psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right) .$$
Show that $\psi _ { j , k } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$.
Q10a Proof Proof That a Map Has a Specific Property View
For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and we set $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Show that $S _ { j } ( f ) \in \mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ and coincides with $f$ at the points $\left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ for $\left( \ell _ { 1 } , \ell _ { 2 } \right) \in \mathbb { Z } ^ { 2 }$.
Q10b Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Let $j \geqslant 2 , k _ { 1 }$ and $k _ { 2 }$ be two integers such that $0 \leqslant k _ { 1 } , k _ { 2 } < j$, and $$\theta \in \left[ \frac { k _ { 1 } } { j } , \frac { k _ { 1 } + 1 } { j } \left[ \times \left[ \frac { k _ { 2 } } { j } , \frac { k _ { 2 } + 1 } { j } [ . \right. \right. \right.$$
Express $S _ { j } ( f ) ( \theta )$ as a barycenter of the points $f \left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ where $\ell _ { 1 } \in \left\{ k _ { 1 } , k _ { 1 } + 1 \right\}$ and $\ell _ { 2 } \in \left\{ k _ { 2 } , k _ { 2 } + 1 \right\}$. Deduce that $\left\| S _ { j } ( f ) - f \right\| _ { \infty } \rightarrow 0$ when $j \rightarrow + \infty$.
Q11 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Using the results of questions 8, 9, 10a and 10b, conclude that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$.
Q12 Differential equations Solving Separable DEs with Initial Conditions View
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 } , x \in \mathbb { R }$ two parameters. We consider the following problem $$\left\{ \begin{array} { l } F ^ { \prime } ( t ) = f ( \alpha ( t ) ) \\ \alpha ^ { \prime } ( t ) = \omega + x g ( \alpha ( t ) ) \end{array} \right.$$ with the initial conditions $F ( 0 ) = 0$ and $\alpha ( 0 ) = ( 0,0 )$, where $\alpha : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$ and $F : \mathbb { R } \rightarrow \mathbb { C }$ are the unknown functions. We assume that $f$ has zero average, that is $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$.
We assume $x = 0$. Determine the unique solution $( F , \alpha )$ of system (3) with initial conditions $F ( 0 ) = 0$ and $\alpha ( 0 ) = ( 0,0 )$.
Q13 Differential equations Solving Separable DEs with Initial Conditions View
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We consider the system $$\left\{ \begin{array} { l } F ^ { \prime } ( t ) = f ( \alpha ( t ) ) \\ \alpha ^ { \prime } ( t ) = \omega \end{array} \right.$$ with $F(0)=0$, $\alpha(0)=(0,0)$. We assume that $f$ has zero average. The vector $\omega = \left( \omega _ { 1 } , \omega _ { 2 } \right)$ is said to be resonant if there exists $\left( k _ { 1 } , k _ { 2 } \right) \in \mathbb { Z } ^ { 2 } \backslash \{ ( 0,0 ) \}$ such that $k _ { 1 } \omega _ { 1 } + k _ { 2 } \omega _ { 2 } = 0$.
Show that, if $\omega$ is resonant, there exists a function $f$ for which $F ( t ) = t$.
Q14a Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that, if $f$ is a trigonometric polynomial, then $F$ is bounded on $\mathbb { R }$.
Q14b Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that more generally, if $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$, then $F ( t ) = o ( t )$ when $t \rightarrow + \infty$.
Q15a Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We assume $x \neq 0$ (but close to 0). We consider a new unknown function $\tilde { \alpha } : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$, of the form $$\tilde { \alpha } ( t ) = \alpha ( t ) + x h ( \alpha ( t ) ) ,$$ where $h : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is an auxiliary function, 1-periodic in each of its arguments, of class $\mathscr { C } ^ { 1 }$ and of zero average, and which moreover, for some $\nu \in \mathbb { R } ^ { 2 }$, satisfies the equation $$\forall \theta \in \mathbb { R } ^ { 2 } , \quad d h ( \theta ) \cdot \omega + g ( \theta ) = \nu . \tag{4}$$
Determine $\nu$ as a function of $g$. In the case where the two components $g _ { 1 }$ and $g _ { 2 }$ of $g$ are trigonometric polynomials, deduce the existence of a solution $h$ of equation (4), which you will make explicit.
Q15b Differential equations Convergence and Approximation of DE Solutions View
We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$.
Show that there exists a function $\varepsilon : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\tilde { \alpha } ^ { \prime } ( t ) = \omega + x \nu + x \varepsilon ( x , t )$$ and $\sup _ { t \in \mathbb { R } } \| \varepsilon ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.
Q15c Differential equations Convergence and Approximation of DE Solutions View
We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$.
Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$ and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.