The function $f$ is defined by the power series $$f(x) = -\frac{x}{2} + \frac{2x^{2}}{3} - \frac{3x^{3}}{4} + \cdots + \frac{(-1)^{n} n x^{n}}{n+1} + \cdots$$ for all real numbers $x$ for which the series converges. The function $g$ is defined by the power series $$g(x) = 1 - \frac{x}{2!} + \frac{x^{2}}{4!} - \frac{x^{3}}{6!} + \cdots + \frac{(-1)^{n} x^{n}}{(2n)!} + \cdots$$ for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The graph of $y = f(x) - g(x)$ passes through the point $(0, -1)$. Find $y'(0)$ and $y''(0)$. Determine whether $y$ has a relative minimum, a relative maximum, or neither at $x = 0$. Give a reason for your answer.

The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$.
(b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$.
(c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.

(Candidates who have followed the specialisation course)
We define the sequence of real numbers $(a_{n})$ by: $$\begin{cases} a_{0} & = 0 \\ a_{1} & = 1 \\ a_{n+1} & = a_{n} + a_{n-1} \text{ for } n \geqslant 1 \end{cases}$$ This sequence is called the Fibonacci sequence.

1. Copy and complete the algorithm below so that at the end of its execution the variable $A$ contains the term $a_{n}$. \begin{verbatim} $A \leftarrow 0$ $B \leftarrow 1$ For $i$ going from 1 to $n$ : $C \leftarrow A + B$ $A \leftarrow \ldots$ $B \leftarrow \ldots$ End For \end{verbatim} We thus obtain the first values of the sequence $a_{n}$:
   

   $n$	0	1	2	3	4	5	6	7	8	9	10
   $a_{n}$	0	1	1	2	3	5	8	13	21	34	55

   
   
2. Let the matrix $A = \left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right)$.
   Calculate $A^{2}$, $A^{3}$ and $A^{4}$. Verify that $A^{5} = \left(\begin{array}{ll} 8 & 5 \\ 5 & 3 \end{array}\right)$.
3. We can prove, and we will admit, that for every non-zero natural number $n$, $$A^{n} = \left(\begin{array}{cc} a_{n+1} & a_{n} \\ a_{n} & a_{n-1} \end{array}\right)$$ a. Let $p$ and $q$ be two non-zero natural numbers. Calculate the product $A^{p} \times A^{q}$ and deduce that $$a_{p+q} = a_{p} \times a_{q+1} + a_{p-1} \times a_{q}$$ b. Deduce that if an integer $r$ divides the integers $a_{p}$ and $a_{q}$, then $r$ also divides $a_{p+q}$.

Let $\mathbb { Z } ^ { + }$ denote the set of positive integers. We want to find all functions $g : \mathbb { Z } ^ { + } \rightarrow \mathbb { Z } ^ { + }$ such that the following equation holds for any $m, n$ in $\mathbb { Z } ^ { + }$. $$g ( n + m ) = g ( n ) + n m ( n + m ) + g ( m )$$ Prove that $g ( n )$ must be of the form $\sum _ { i = 0 } ^ { d } c _ { i } n ^ { i }$ and find the precise necessary and sufficient condition(s) on $d$ and on the coefficients $c _ { 0 } , \ldots , c _ { d }$ for $g$ to satisfy the required equation.

For a natural number $k$, when $n = 5 ^ { k }$, $f ( n )$ satisfies $$f ( 5 n ) = f ( n ) + 3 , \quad f ( 5 ) = 4$$ Find the value of $\sum _ { k = 1 } ^ { 10 } f \left( 5 ^ { k } \right)$. [4 points]

A square is divided into three equal parts horizontally to create [Figure 1], and divided into three equal parts vertically to create [Figure 2]. [Figure 1] and [Figure 2] are alternately attached repeatedly to create the following figure. As shown in the figure, let A be the upper left vertex of the first attached [Figure 1], and let $\mathrm { B } _ { n }$ be the lower right vertex of the figure created by attaching a total of $n$ figures (combining the number of [Figure 1] and [Figure 2]).
When $a _ { n }$ is the number of shortest paths from vertex A to vertex $\mathrm { B } _ { n }$ along the lines, what is the value of $a _ { 3 } + a _ { 7 }$? [4 points]
(1) 26
(2) 28
(3) 30
(4) 32
(5) 34

Points are marked on the coordinate plane in the following [Steps]. [Step 1] Mark a point at $( 0,1 )$. [Step 2] Mark 3 points at $( 0,3 ) , ( 1,3 ) , ( 2,3 )$ in this order. $\vdots$ [Step $k$ ] Mark $( 2 k - 1 )$ points at $( 0,2 k - 1 ) , ( 1,2 k - 1 ) , ( 2,2 k - 1 ) , \cdots$, $( 2 k - 2,2 k - 1 )$ in this order. (Here, $k$ is a natural number.) $\vdots$ When points are marked in this manner starting from [Step 1], the coordinates of the 100th marked point are $( p , q )$. What is the value of $p + q$? [4 points]
(1) 46
(2) 43
(3) 40
(4) 37
(5) 34

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text{ is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$. [4 points]

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text { is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$.

21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$. (I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$; (II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.

Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.

The answer to a problem is: The sequence is $1, 1, 2, 2^0, 2^1$, and the next three terms are $2^0, 2^1, 2^2$, and so on. The first term is $2^0$. Find the smallest positive integer $N$ such that $N > 100$ and the sum of the first $N$ terms of this sequence is an integer power of 2.
A. 440
B. 330
C. 220
D. 110

After completing its lunar exploration mission, the Chang'e-2 satellite continued deep space exploration and became China's first artificial planet orbiting the sun. To study the ratio of Chang'e-2's orbital period around the sun to Earth's orbital period around the sun, the sequence $\{b_n\}$ is used: $b_1 = 1 + \frac{1}{a_1}, b_2 = 1 + \frac{1}{a_1 + \frac{1}{a_2}}, b_3 = 1 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}, \cdots$, and so on, where $a_k \in \mathbf{N}^* (k = 1,2,\cdots)$. Then
A. $b_1 < b_5$
B. $b_3 < b_8$
C. $b_6 < b_2$
D. $b_4 < b_7$

Let $f$ be a real function, defined, continuous and decreasing on $[ a , + \infty [$, where $a \in \mathbb { R }$. Show that for every integer $k \in \left[ a + 1 , + \infty \left[ \right. \right.$, we have $\int _ { k } ^ { k + 1 } f ( x ) d x \leqslant f ( k ) \leqslant \int _ { k - 1 } ^ { k } f ( x ) d x$.

Deduce the nature of the Riemann series $\sum _ { n \geqslant 1 } \frac { 1 } { n ^ { \alpha } }$ according to the value of $\alpha \in \mathbb { R }$.

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Deduce that $$R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + \frac { 1 } { 2 n ^ { \alpha } } + O \left( \frac { 1 } { n ^ { \alpha + 1 } } \right)$$

Show that there exists a real sequence $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ having the following property: for every integer $p \in \mathbb { N } ^ { * }$, for every non-degenerate interval $I$ and for every complex function $f$ of class $C ^ { \infty }$ on $I$, the function $g$ defined on $I$ by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { p - 1 } f ^ { ( p - 1 ) }$ satisfies $$g ^ { \prime } + \frac { 1 } { 2 ! } g ^ { \prime \prime } + \frac { 1 } { 3 ! } g ^ { ( 3 ) } + \cdots + \frac { 1 } { p ! } g ^ { ( p ) } = f ^ { \prime } + \sum _ { l = 1 } ^ { p - 1 } b _ { l , p } f ^ { ( p + l ) }$$ where the $b _ { l , p }$ are coefficients independent of $f$ which we do not seek to calculate.

Show that $a _ { 0 } = 1$ and that for every $p \geqslant 1 , a _ { p } = - \sum _ { i = 2 } ^ { p + 1 } \frac { a _ { p + 1 - i } } { i ! }$. Deduce that $\left| a _ { p } \right| \leqslant 1$ for every natural integer $p$. Determine $a _ { 1 }$ and $a _ { 2 }$.

a) For every $z \in \mathbb { C }$ such that $| z | < 1$, justify that the series $\sum _ { p \in \mathbb { N } } a _ { p } z ^ { p }$ is convergent.
We denote by $\varphi ( z )$ its sum: $\varphi ( z ) = \sum _ { p = 0 } ^ { \infty } a _ { p } z ^ { p }$.
b) For $z \in \mathbb { C }$ such that $| z | < 1$, calculate the product $\left( e ^ { z } - 1 \right) \varphi ( z )$. Deduce that for every $z \in \mathbb { C } ^ { * }$ satisfying $| z | < 1$, we have $\varphi ( z ) = \frac { z } { e ^ { z } - 1 }$.
c) Show that $a _ { 2 k + 1 } = 0$ for every integer $k \geqslant 1$. Calculate $a _ { 4 }$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$.
Deduce the asymptotic expansion of the remainder $$R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } } = - \left( a _ { 0 } f ( n ) + a _ { 1 } f ^ { \prime } ( n ) + a _ { 2 } f ^ { \prime \prime } ( n ) + \cdots + a _ { 2 p - 2 } f ^ { ( 2 p - 2 ) } ( n ) \right) + O \left( \frac { 1 } { n ^ { 2 p + \alpha - 1 } } \right)$$

Give the asymptotic expansion of $R _ { n } ( 3 )$ corresponding to the case $\alpha = 3$ and $p = 3$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } } = \left( A _ { n } ( X ) \right) _ { n \in \mathbb { N } }$ satisfying the following conditions $$A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n } \text { and } \int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0 \text { for every } n \in \mathbb { N }$$ The polynomials $B _ { n } = n ! A _ { n }$ are called Bernoulli polynomials.
a) Show that the sequence $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ is uniquely determined by the conditions above; specify the degree of $A _ { n }$; calculate $A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$.
b) Show that $A _ { n } ( t ) = ( - 1 ) ^ { n } A _ { n } ( 1 - t )$ for every $n \in \mathbb { N }$ and every $t \in \mathbb { R }$.
c) For every integer $n \geqslant 2$, show that $A _ { n } ( 0 ) = A _ { n } ( 1 )$ and that $A _ { 2 n - 1 } ( 0 ) = 0$.
d) We provisionally set $c _ { n } = A _ { n } ( 0 )$ for every natural integer $n$. Show that for every $n \in \mathbb { N }$, $$A _ { n } ( X ) = c _ { 0 } \frac { X ^ { n } } { n ! } + \cdots + c _ { n - 2 } \frac { X ^ { 2 } } { 2 ! } + c _ { n - 1 } X + c _ { n }$$ then that if $n \geqslant 1$, $$\frac { c _ { 0 } } { ( n + 1 ) ! } + \cdots + \frac { c _ { n - 2 } } { 3 ! } + \frac { c _ { n - 1 } } { 2 ! } + c _ { n } = 0$$
e) Deduce that for every $n \in \mathbb { N }$, we actually have $c _ { n } = a _ { n }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$.
a) Show that the series $\sum _ { n } A _ { n } ( t ) z ^ { n }$ converges for every real $t \in [ - 1,1 ]$ and every complex $z$ satisfying $| z | < 1$.
Under these conditions, we set $f ( t , z ) = \sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n }$.
b) Let $z \in \mathbb { C }$ such that $| z | < 1$. Show that the function $t \mapsto f ( t , z )$ is differentiable on $[ 0,1 ]$ and express its derivative in terms of $f ( t , z )$. Deduce that if $| z | < 1$ and $z \neq 0$, $$\sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n } = \frac { z e ^ { t z } } { e ^ { z } - 1 }$$
c) Show that if $z \in \mathbb { C }$ and $| z | < 2 \pi$, we have $\frac { z e ^ { z / 2 } } { e ^ { z } - 1 } + \frac { z } { e ^ { z } - 1 } = 2 \frac { z / 2 } { e ^ { z / 2 } - 1 }$.
Deduce that for every natural integer $n , A _ { n } \left( \frac { 1 } { 2 } \right) = \left( \frac { 1 } { 2 ^ { n - 1 } } - 1 \right) a _ { n }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$. For $n \geqslant 2$, the variations of $A_n$ on $[0,1]$ are known, in particular:

1. If $n \equiv 2 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) > 0 > A _ { n } \left( \frac { 1 } { 2 } \right)$.
2. If $n \equiv 0 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } ( 1 ) < 0 < A _ { n } \left( \frac { 1 } { 2 } \right)$.
3. If $n \equiv 1 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } < 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } > 0$ on $]\frac{1}{2} , 1[$.
4. If $n \equiv 3 \bmod 4$, then $A _ { n } ( 0 ) = A _ { n } \left( \frac { 1 } { 2 } \right) = A _ { n } ( 1 ) = 0$; $A _ { n } > 0$ on $]0 , \frac{1}{2}[$ and $A _ { n } < 0$ on $]\frac{1}{2} , 1[$.

a) Show that for $n \geqslant 2$, the variations of the polynomials $A _ { n }$ on $[ 0,1 ]$ correspond to the four cases described above.
b) For every $n \in \mathbb { N } ^ { * }$ and every $x \in [ 0,1 ]$, show that $\left| A _ { 2 n } ( x ) \right| \leqslant \left| a _ { 2 n } \right|$ and $\left| A _ { 2 n + 1 } ( x ) \right| \leqslant \frac { \left| a _ { 2 n } \right| } { 2 }$.

Let $f$ be a complex function of class $C ^ { \infty }$ on $[ 0,1 ]$.
a) Show that for every integer $q \geqslant 1$ $$f ( 1 ) - f ( 0 ) = \sum _ { j = 1 } ^ { q } ( - 1 ) ^ { j + 1 } \left[ A _ { j } ( t ) f ^ { ( j ) } ( t ) \right] _ { 0 } ^ { 1 } + ( - 1 ) ^ { q } \int _ { 0 } ^ { 1 } A _ { q } ( t ) f ^ { ( q + 1 ) } ( t ) d t$$
b) Taking into account the relations found in the previous part, show that for every odd natural integer $q = 2 p + 1$ $$f ( 1 ) - f ( 0 ) = \frac { 1 } { 2 } \left( f ^ { \prime } ( 0 ) + f ^ { \prime } ( 1 ) \right) - \sum _ { j = 1 } ^ { p } a _ { 2 j } \left( f ^ { ( 2 j ) } ( 1 ) - f ^ { ( 2 j ) } ( 0 ) \right) - \int _ { 0 } ^ { 1 } A _ { 2 p + 1 } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Justify the convergence of the series $\sum_{k \geqslant 2} w_{k}$.
Deduce that there exists a real number $a$ such that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + v_{n}$$ where $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.