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 $$\forall x \in ]-R, R[, \quad S(x) = W(x).$$

Let $\left(u_n\right)_{n \in \mathbb{N}}$ be a hypergeometric sequence with associated polynomials $P$ and $Q$. Suppose that there exists a natural integer $n_0$ such that $P\left(n_0\right) = 0$ and, $\forall n \geqslant n_0, Q(n) \neq 0$. Justify that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is zero from a certain rank onwards.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\varepsilon$ is continuous on $I \backslash \left\{ \frac { 1 } { 4 } \right\}$. Deduce $$\forall t \in I , \quad g ( t ) = 1 - \sqrt { 1 - 4 t } .$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that, for all $x \in ]0, +\infty[$, we have $\Gamma(x+1) = x\Gamma(x)$.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that $\lim_{x \rightarrow +\infty} (\ln(\Gamma))''(x) = 0$.

We set $r_\infty := \lim r_k$ and $$h_k := (I + P_0) \circ (I + P_1) \circ \cdots \circ (I + P_{k-1}).$$ Explain why $r_\infty$ is well defined, and show that $\hat{h}_k(r_k) \leqslant r_0$ for all $k \geqslant 1$. Deduce that the series $h$ of question E satisfies $\hat{h}(r_\infty) \leqslant r_0$, thus that $\rho(h) \geqslant r_\infty$.

Let $\ell$ be a strictly positive integer. We are given a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ of vectors in $\mathbb { R } ^ { \ell }$ such that the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$ converges.
(a) Show that the sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ is convergent.
(b) Let $v ^ { * }$ denote the limit of this sequence. Bound $\left\| v _ { n } - v ^ { * } \right\|$ by means of a remainder of the sum of the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$.
15a. Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$.
15b. Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.

Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability. We define a random variable $Z_{n}$ by $Z_{n}(\sigma) = \nu(\sigma)$.
Calculate, for any natural integer $k \leqslant n$, $\lim_{n \rightarrow +\infty} \mathbb{P}\left(Z_{n} = k\right)$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Calculate, for any natural integer $k \leqslant n$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Z_n = k)$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \right)$$ where $\sigma_n = \frac{1}{n+1}\sum_{k=0}^n u_k$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ takes real values, prove that the result holds if $\ell = + \infty$ or $\ell = - \infty$.

Using (Cesàro), calculate the limit of the sequence $\left( v _ { n } \right) _ { n \geqslant 1 }$ defined by $v _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k n }$. Then, using a series-integral comparison, give an equivalent of $\left( v _ { n } \right) _ { n \geqslant 1 }$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ and $\alpha \in \mathbb { R } ^ { * }$. Suppose that $\lim _ { n \rightarrow + \infty } e _ { n } = \alpha$, where $e_n = u_{n+1} - u_n$. Using (Cesàro), give an equivalent of $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Recover this result using a comparison theorem for series with positive terms.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \left] 0 , + \infty \right[ ^ { \mathbb { N } }$ and $\ell \in \left] 0 , + \infty \right[$. Suppose that $\lim _ { n \rightarrow + \infty } \frac { u _ { n + 1 } } { u _ { n } } = \ell$. Prove that $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { u _ { n } } = \ell$. Prove that the result holds if $\ell = 0$ or $\ell = + \infty$. Deduce $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { n ! }$ and $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { \frac { n ^ { n } } { n ! } }$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } } , \left( b _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } } , a \in \mathbb { C }$ and $b \in \mathbb { C }$. Suppose that $\lim _ { n \rightarrow + \infty } a _ { n } = a$ and $\lim _ { n \rightarrow + \infty } b _ { n } = b$. Prove that $$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } \right) = a b$$

Let $\sum _ { n \geqslant 0 } a _ { n }$ and $\sum _ { n \geqslant 0 } b _ { n }$ be two series of complex numbers, convergent with respective sums $A$ and $B$. We denote $\left( c _ { n } \right) _ { n \in \mathbb { N } }$ the sequence with general term $c _ { n } = \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k }$ and $\left( C _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of partial sums associated defined by $C _ { n } = \sum _ { k = 0 } ^ { n } c _ { k }$. Prove that $$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } C _ { k } \right) = A B \qquad \text{(Cauchy)}$$

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ and $\ell \in \mathbb { R }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } \left( u _ { n } \right) _ { n \in \mathbb { N } } \text { monotone } \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) .$$ Prove that the result holds for $\ell = + \infty$ or $\ell = - \infty$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Weak Hardy)}$$ Hint: one may prove that for all $n \geqslant 1$, $$\sum _ { k = 0 } ^ { n } k e _ { k } = n u _ { n + 1 } - \sum _ { k = 1 } ^ { n } u _ { k }$$

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = O \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Strong Hardy)}$$
We suppose that $\lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell$ and $e _ { n } = O \left( \frac { 1 } { n } \right)$.
(a) Let $0 \leqslant n < m$. Prove that $$\sum _ { k = n + 1 } ^ { m } u _ { k } - ( m - n ) u _ { n } = \sum _ { j = n } ^ { m - 1 } ( m - j ) e _ { j }$$
(b) Deduce that there exists a constant $C > 0$ such that for all $2 \leqslant n < m$, we have $$\left| \frac { ( m + 1 ) \sigma _ { m } - ( n + 1 ) \sigma _ { n } } { m - n } - u _ { n } \right| \leqslant C \ln \left( \frac { m - 1 } { n - 1 } \right)$$ and $$\left| u _ { n } - \ell \right| \leqslant C \ln \left( \frac { m - 1 } { n - 1 } \right) + \frac { m + 1 } { m - n } \left( \left| \sigma _ { m } - \ell \right| + \left| \sigma _ { n } - \ell \right| \right) .$$
(c) Deduce (Strong Hardy). Hint: one may take $m = 1 + \lfloor \alpha n \rfloor$ with a parameter $\alpha > 1$ to be chosen, where $\lfloor x \rfloor$ denotes the integer part of $x \in \mathbb { R }$.

Express $f ( 0 )$ and $\lim _ { x \rightarrow + \infty } f ( x )$ in terms of $a _ { 0 }$ and $b _ { 0 }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, $\lambda_0 = 0$, and $b_0 = \sum_{n=1}^{+\infty} a_n$.

Suppose that $y$ is the sum of a Dirichlet series: $$\forall x \in \mathbf { R } _ { + } \quad y ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x },$$ where $y(0) = 0$ and $\lim_{x\to+\infty} y(x) = c$. Express $a _ { 0 }$ and $b _ { 0 }$ in terms of the constant $c$ introduced in Part I.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
Suppose in this question that $b = 0$. Show that $(u _ { n })$ tends to 0.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
In the general case, show that $(u _ { n })$ converges and specify its limit.

Problem 2, Part 3: Linear recurrence sequences with variable coefficients
We consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form $$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$ where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \} , \left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0 , V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied.
Deduce that $v _ { n }$ tends to 0.