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.

9. Suppose in this question that $b = 0$. Show that $( u _ { n } )$ tends to 0.

10. In the general case, show that $( u _ { n } )$ converges and specify its limit.

Part 3: Linear recurrent sequences with variable coefficients

We keep the notation from the previous part and we now 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.

113. For values $n \geq n_0$, if the distance of the terms of the sequence $\left\{\dfrac{fn+1}{rn-2}\right\}$ from its limit is less than $0.02$, what is the smallest value of $n_0$?
(1) $61$ (2) $62$ (3) $63$ (4) $64$

113- For natural numbers $n \geq n_0$, the sequence $\left\{\dfrac{2n^2+1}{n^2+2n}\right\}$ converges to its limit point, with a distance less than $0.04$. The smallest value of $n_0$ is which of the following?
(1) $96$ (2) $97$ (3) $98$ (4) $99$

115. The sequence $\{x_n\}$ is defined as follows. What is the limit of $\{x_n\}$?
$$x_0 = 3 \;,\quad x_{n+1} = \frac{3x_n^2 + 64}{4x_n^2} \;,\quad (n = 1, 2, \ldots)$$
$$2\sqrt{2} \quad (1) \qquad -2\sqrt{2} \quad (2) \qquad 2\sqrt[4]{2} \quad (3) \qquad -2\sqrt[4]{2} \quad (4)$$

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 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 $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that
$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$
Then which of the following is (are) correct?
(A) $( - 1,3 ) \in S$
(B) $( - 1,1 ) \in S$
(C) $( 1 , - 1 ) \in S$
(D) $( 1 , - 2 ) \in S$

The value of $4 + \frac { 1 } { 5 + \frac { 1 } { 4 + \frac { 1 } { 5 + \frac { 1 } { 4 + \ldots . . \infty } } } }$ is:
(1) $2 + \frac { 2 } { 5 } \sqrt { 30 }$
(2) $2 + \frac { 4 } { \sqrt { 5 } } \sqrt { 30 }$
(3) $4 + \frac { 4 } { \sqrt { 5 } } \sqrt { 30 }$
(4) $5 + \frac { 2 } { 5 } \sqrt { 30 }$

The value of $3 + \frac { 1 } { 4 + \frac { 1 } { 3 + \frac { 1 } { 4 + \frac { 1 } { 3 + \ldots . \infty } } } }$ is equal to
(1) $1.5 + \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $3 + 2 \sqrt { 3 }$
(4) $4 + \sqrt { 3 }$

Let $\beta = \lim _ { x \rightarrow 0 } \frac { \alpha x - \left( e ^ { 3 x } - 1 \right) } { \alpha x \left( e ^ { 3 x } - 1 \right) }$ for some $\alpha \in \mathbb { R }$. Then the value of $\alpha + \beta$ is:
(1) $\frac { 14 } { 5 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 1 } { 2 }$

If $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } - n - 1 } + n\alpha + \beta \right) = 0$ then $8\alpha + \beta$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 8

If $\lim _ { n \rightarrow \infty } \frac { (n+1)^{k-1} } { n ^ { k + 1 } } \left[ (nk+1) + (nk+2) + \ldots + (nk+n) \right] = 33 \cdot \lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { k + 1 } } \cdot \left( 1 ^ { k } + 2 ^ { k } + 3 ^ { k } + \ldots + n ^ { k } \right)$, then the integral value of $k$ is equal to $\_\_\_\_$.

$\lim _ { t \rightarrow 0 } 1 ^ { \frac { 1 } { \sin ^ { 2 } t } } + 2 ^ { \frac { 1 } { \sin ^ { 2 } t } } + 3 ^ { \frac { 1 } { \sin ^ { 2 } t } } \ldots \ldots n ^ { \frac { 1 } { \sin ^ { 2 } t } } \sin ^ { 2 } t$ is equal to
(1) $n ^ { 2 } + n$
(2) $n$
(3) $\frac { n n + 1 } { 2 }$
(4) $n ^ { 2 }$

$\lim_{n \rightarrow \infty} \left\{\left(2^{\frac{1}{2}} - 2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}} - 2^{\frac{1}{5}}\right) \ldots \left(2^{\frac{1}{2}} - 2^{\frac{1}{2n+1}}\right)\right\}$ is equal to
(1) 1
(2) 0
(3) $\sqrt{2}$
(4) $\frac{1}{\sqrt{2}}$

The value of $\lim _ { n \rightarrow \infty } \frac { 1 + 2 - 3 + 4 + 5 - 6 + \ldots + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n } { \sqrt { 2 n ^ { 4 } + 4 n + 3 } - \sqrt { n ^ { 4 } + 5 n + 4 } }$ is
(1) $\frac { \sqrt { 2 } + 1 } { 2 }$
(2) $3 ( \sqrt { 2 } + 1 )$
(3) $\frac { 3 } { 2 } ( \sqrt { 2 } + 1 )$
(4) $\frac { 3 } { 2 \sqrt { 2 } }$

$\lim _ { n \rightarrow \infty } \frac { \left( 1 ^ { 2 } - 1 \right) ( n - 1 ) + \left( 2 ^ { 2 } - 2 \right) ( n - 2 ) + \cdots + \left( ( n - 1 ) ^ { 2 } - ( n - 1 ) \right) \cdot 1 } { \left( 1 ^ { 3 } + 2 ^ { 3 } + \cdots \cdots + n ^ { 3 } \right) - \left( 1 ^ { 2 } + 2 ^ { 2 } + \cdots \cdots + n ^ { 2 } \right) }$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 1 } { 2 }$

Q68. $\lim _ { n \rightarrow \infty } \frac { \left( 1 ^ { 2 } - 1 \right) ( n - 1 ) + \left( 2 ^ { 2 } - 2 \right) ( n - 2 ) + \cdots + \left( ( n - 1 ) ^ { 2 } - ( n - 1 ) \right) \cdot 1 } { \left( 1 ^ { 3 } + 2 ^ { 3 } + \cdots \cdots + n ^ { 3 } \right) - \left( 1 ^ { 2 } + 2 ^ { 2 } + \cdots \cdots + n ^ { 2 } \right) }$ is equal to :
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 1 } { 2 }$

Suppose two sequences $\langle a_n \rangle$ and $\langle b_n \rangle$ satisfy $b_n + \frac{4n-1}{n} < a_n < 3b_n$ for all positive integers $n$. Given that $\lim_{n \to \infty} a_n = 6$, select the correct options.
(1) $b_n < 6 - \frac{4n-1}{n}$
(2) $b_n > \frac{4n-1}{2n}$
(3) The sequence $\langle b_n \rangle$ may diverge
(4) $a_{10000} < 6.1$
(5) $a_{10000} > 5.9$