grandes-ecoles

Papers (176)

2025

centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30

2024

centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23

2023

centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22

2022

centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26

2021

centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29

2020

centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6

2019

centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9

2018

centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20

2017

centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19

2016

centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20

2015

centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13

2013

centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20

2011

centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28

2010

centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27

2015 x-ens-maths2__mp

19 maths questions

Q1a The Gamma Distribution 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 The Gamma Distribution 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 Integration with Partial Fractions 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 Integration with Partial Fractions 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 Integration with Partial Fractions 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 Integration with Partial Fractions 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 Differential equations 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 Taylor 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 Sequences and 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 Sequences and Series 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 Sequences and 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 Indefinite & Definite Integrals 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 Taylor series Evaluate a Closed-Form Expression Using the Reduction Formula View

Q6a Taylor 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 Taylor 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 Taylor 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 Taylor 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 Taylor 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 Taylor 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 $E _ { 4 } \left( \frac { 1 } { 10 } \right) \leqslant 3.10 ^ { - 3 }$.