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.

Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$. We suppose in this question that $0 < x_0 < 1/\tau$. a) Justify that the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ is decreasing, with strictly positive values, and satisfies $x_{n+1} = x_n(1 - \tau x_n)$ for all $n \in \mathbb{N}$. b) Justify that $x_n \rightarrow 0$ when $n \rightarrow \infty$. c) Show that $1/x_{n+1} = 1/x_n + \tau/(1 - \tau x_n)$ for all $n \in \mathbb{N}$. Deduce that $x_n \leq x_0/(1 + n\tau x_0)$.

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 (all complex roots of $P(X) = X^d - \sum_{i=0}^{d-1} a_i X^i$ have modulus strictly less than 1), and $A$ is the matrix from question 7.
Let $\varepsilon > 0$ be fixed. Show that there exists an integer $q \geqslant 1$ and an integer $n _ { 0 }$ such that for all $n \geqslant n _ { 0 }$, $$\left\| V _ { n + q } \right\| _ { \infty } \leqslant ( \sigma ( A ) + \varepsilon ) ^ { q } \left\| V _ { n } \right\| _ { \infty }$$ where $A$ is the matrix from question 7.

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.

Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.

Let $f(a, b)$ be a function satisfying $f(a, b) = f(a, c) + f(c, b) - 2f(a,c)f(c,b)$ with $f(99, 100) = 1/3$. Find $f(1, 100)$.

Let $f(n)$ satisfy the recurrence $f(n) + f(n-1) = nf(n-1) + (n-1)f(n-2)$ with $f(0) = 1$, $f(1) = 0$. Find a closed form for $f(n)$.

Let $a_1 = 24^{1/3}$ and $a_{n+1} = (a_n + 24)^{1/3}$. Find the integer part of $a_{100}$.

Find $\lim_{n\to\infty}\left(1 + \dfrac{1}{n}\right)^n$.

Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true?
(A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
(B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$
(C) $x_n \to 1$ for all $a > 0$
(D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{ a_n \}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{a_n\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function such that its derivative $f ^ { \prime }$ is a continuous function. Moreover, assume that for all $x \in \mathbb { R }$, $$0 \leq \left| f ^ { \prime } ( x ) \right| \leq \frac { 1 } { 2 }$$ Define a sequence of real numbers $\left\{ a _ { n } \right\} _ { n \in \mathbb { N } }$ by: $$\begin{gathered} a _ { 1 } = 1 , \\ a _ { n + 1 } = f \left( a _ { n } \right) \text { for all } n \in \mathbb { N } . \end{gathered}$$ Prove that there exists a positive real number $M$ such that for all $n \in \mathbb { N }$, $$\left| a _ { n } \right| \leq M$$

Consider the real-valued function $h : \{ 0,1,2 , \ldots , 100 \} \rightarrow \mathbb { R }$ such that $h ( 0 ) = 5 , h ( 100 ) = 20$ and satisfying $h ( i ) = \frac { 1 } { 2 } ( h ( i + 1 ) + h ( i - 1 ) )$, for every $i = 1,2 , \ldots , 99$. Then, the value of $h ( 1 )$ is:
(A) 5.15
(B) 5.5
(C) 6
(D) 6.15.

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be defined by $$f(x) = \lim_{n \rightarrow \infty} \cos^{n}\left(\frac{1}{n^{x}}\right)$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.

Let a real-valued sequence $\left\{x_{n}\right\}_{n \geq 1}$ be such that
$$\lim_{n \rightarrow \infty} n x_{n} = 0$$
Find all possible real values of $t$ such that $\lim_{n \rightarrow \infty} x_{n}(\log n)^{t} = 0$.

Let $\mathbb { Z }$ denote the set of integers. Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be such that $f ( x ) f ( y ) = f ( x + y ) + f ( x - y )$ for all $x , y \in \mathbb { Z }$. If $f ( 1 ) = 3$, then $f ( 7 )$ equals
(A) 840
(B) 844
(C) 843
(D) 842

Let $\left\{ u _ { n } \right\} _ { n \geq 1 }$ be a sequence of real numbers defined as $u _ { 1 } = 1$ and
$$u _ { n + 1 } = u _ { n } + \frac { 1 } { u _ { n } } \text { for all } n \geq 1 .$$
Prove that $u _ { n } \leq \frac { 3 \sqrt { n } } { 2 }$ for all $n$.

The limit $$\lim _ { n \rightarrow \infty } \left( 2 ^ { - 2 ^ { n + 1 } } + 2 ^ { - 2 ^ { n - 1 } } \right) ^ { 2 ^ { - n } }$$ equals
(A) 1.
(B) $\frac { 1 } { \sqrt { 2 } }$.
(C) 0.
(D) $\frac { 1 } { 4 }$.