Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote $$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$
Q2
First order differential equations (integrating factor)View
Let $z \in D$. Show that the function $t \in [ 0,1 ] \mapsto L ( t z )$ is differentiable and give a simple expression for its derivative. Deduce that $t \mapsto ( 1 - t z ) e ^ { L ( t z ) }$ is constant on $[ 0,1 ]$ and conclude that $$\exp ( L ( z ) ) = \frac { 1 } { 1 - z }$$
Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$, $$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$
For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is finite for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing, and that it is constant from rank $\max ( n , 1 )$ onward.
Show by induction that $$\forall N \in \mathbf { N } ^ { * } , \forall z \in D , \prod _ { k = 1 } ^ { N } \frac { 1 } { 1 - z ^ { k } } = \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$$
Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that $$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$ Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.
Let $n \in \mathbf { N }$. Show that for all real $t > 0$, $$p _ { n } = \frac { e ^ { n t } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } P \left( e ^ { - t + i \theta } \right) \mathrm { d } \theta$$ so that $$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t + i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$
Let $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that $$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x )$$ Deduce that for all $x \in [ 0,1 [$ and all real $\theta$, $$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$
We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set $$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$ We also introduce the function $$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$ which is obviously of class $\mathcal { C } ^ { \infty }$. Show that $\varphi _ { n , \alpha }$ and $\varphi _ { n , \alpha } ^ { \prime }$ are integrable on $] 0 , + \infty [$.
We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set $$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$ We also introduce the function $$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$ Show, for all real $t > 0$, the existence of $S _ { n , \alpha } ( t )$, its strict positivity, and the identity $$\int _ { 0 } ^ { + \infty } \varphi _ { n , \alpha } ( x ) \mathrm { d } x = t ^ { n + 1 } S _ { n , \alpha } ( t ) - \sum _ { k = 0 } ^ { + \infty } \int _ { k t } ^ { ( k + 1 ) t } ( x - k t ) \varphi _ { n , \alpha } ^ { \prime } ( x ) \mathrm { d } x$$ Deduce that $$S _ { n , \alpha } ( t ) = \frac { 1 } { t ^ { n + 1 } } \int _ { 0 } ^ { + \infty } \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } } \mathrm {~d} x + O \left( \frac { 1 } { t ^ { n } } \right) \quad \text { as } t \rightarrow 0 ^ { + }$$
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Show that for all $( a , b ) \in \mathbf { R } ^ { 2 }$ and all real $\theta$, $$\Phi _ { a X + b } ( \theta ) = \frac { p e ^ { i ( a + b ) \theta } } { 1 - q e ^ { i a \theta } }$$
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Show that for all $k \in \mathbf { N }$, the random variable $X ^ { k }$ has finite expectation. Show that $\Phi _ { X }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbf { R }$ and that $\Phi _ { X } ^ { ( k ) } ( 0 ) = i ^ { k } \mathbf { E } \left( X ^ { k } \right)$ for all $k \in \mathbf { N }$.
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that $$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that $$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Deduce that there exists a real $K > 0$ independent of $p$ such that $$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation. Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that $$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation. Show, for all real $u$, the inequality $$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$ Deduce that for all real $\theta$, $$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$
Let $n \in \mathbf { N } ^ { * }$ as well as complex numbers $z _ { 1 } , \ldots , z _ { n } , u _ { 1 } , \ldots , u _ { n }$ all of modulus at most 1. Show that $$\left| \prod _ { k = 1 } ^ { n } z _ { k } - \prod _ { k = 1 } ^ { n } u _ { k } \right| \leq \sum _ { k = 1 } ^ { n } \left| z _ { k } - u _ { k } \right|$$
Given a real $t > 0$, we set $$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$ Given reals $t > 0$ and $u$, we set $$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$ Show that $\sigma _ { t } \sim \frac { \pi } { \sqrt { 3 } t ^ { 3 / 2 } }$ as $t$ tends to $0 ^ { + }$. Deduce from this that, for all real $u$, $$j ( t , u ) \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } e ^ { - u ^ { 2 } / 2 }$$
Show that there exists a real $\alpha > 0$ such that $$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$ Using question $9 \triangleright$, deduce from this that there exist three real numbers $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $\left. t \in ] 0 , t _ { 0 } \right]$ and all $\theta \in [ - \pi , \pi ]$, $$| h ( t , \theta ) | \leq e ^ { - \beta \left( \sigma _ { t } \theta \right) ^ { 2 } } \quad \text { or } \quad | h ( t , \theta ) | \leq e ^ { - \gamma \left( \sigma _ { t } | \theta | \right) ^ { 2 / 3 } }$$
Given reals $t > 0$ and $u$, we set $$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$ Conclude that $$\int _ { - \pi \sigma _ { t } } ^ { \pi \sigma _ { t } } j ( t , u ) \mathrm { d } u \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } \sqrt { 2 \pi }$$