A function $f$ is defined by $$f(x) = \frac{1}{3} + \frac{2}{3^2}x + \frac{3}{3^3}x^2 + \cdots + \frac{n+1}{3^{n+1}}x^n + \cdots$$ for all $x$ in the interval of convergence of the given power series.
(a) Find the interval of convergence for this power series. Show the work that leads to your answer.
(b) Find $\displaystyle\lim_{x \rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}$.
(c) Write the first three nonzero terms and the general term for an infinite series that represents $\displaystyle\int_0^1 f(x)\, dx$.
(d) Find the sum of the series determined in part (c).

The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is $$\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{2n+3} = \frac{x}{3} - \frac{x^{3}}{5} + \frac{x^{5}}{7} - \cdots$$
(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 $g'(x)$.

Which of the following series converge?
I. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } } { n ! }$
II. $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { 100 } }$
III. $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { ( n ) ( n + 2 ) ( n + 3 ) }$
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

What is the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { 2 n } } { 3 ^ { n } }$ ?
(A) $2 \sqrt { 3 }$
(B) 3
(C) $\sqrt { 3 }$
(D) $\frac { \sqrt { 3 } } { 2 }$
(E) 0

The power series $\sum _ { n = 0 } ^ { \infty } a _ { n } ( x - 3 ) ^ { n }$ converges at $x = 5$. Which of the following must be true?
(A) The series diverges at $x = 0$.
(B) The series diverges at $x = 1$.
(C) The series converges at $x = 1$.
(D) The series converges at $x = 2$.
(E) The series converges at $x = 6$.

For what values of $p$ will both series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 p } }$ and $\sum _ { n = 1 } ^ { \infty } \left( \frac { p } { 2 } \right) ^ { n }$ converge?
(A) $- 2 < p < 2$ only
(B) $- \frac { 1 } { 2 } < p < \frac { 1 } { 2 }$ only
(C) $\frac { 1 } { 2 } < p < 2$ only
(D) $p < \frac { 1 } { 2 }$ and $p > 2$
(E) There are no such values of $p$.

The Taylor series for a function $f$ about $x = 1$ is given by $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n } ( x - 1 ) ^ { n }$ and converges to $f ( x )$ for $| x - 1 | < R$, where $R$ is the radius of convergence of the Taylor series.
(a) Find the value of $R$.
(b) Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 1$.
(c) The Taylor series for $f ^ { \prime }$ about $x = 1$, found in part (b), is a geometric series. Find the function $f ^ { \prime }$ to which the series converges for $| x - 1 | < R$. Use this function to determine $f$ for $| x - 1 | < R$.

The Maclaurin series for a function $f$ is given by $$\sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } = x - \frac { 3 } { 2 } x ^ { 2 } + 3 x ^ { 3 } - \cdots + \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } + \cdots$$ and converges to $f ( x )$ for $| x | < R$, where $R$ is the radius of convergence of the Maclaurin series.
(a) Use the ratio test to find $R$.
(b) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime }$, the derivative of $f$. Express $f ^ { \prime }$ as a rational function for $| x | < R$.
(c) Write the first four nonzero terms of the Maclaurin series for $e ^ { x }$. Use the Maclaurin series for $e ^ { x }$ to write the third-degree Taylor polynomial for $g ( x ) = e ^ { x } f ( x )$ about $x = 0$.

The function $f$ has a Taylor series about $x = 1$ that converges to $f ( x )$ for all $x$ in the interval of convergence. It is known that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - \frac { 1 } { 2 }$, and the $n$th derivative of $f$ at $x = 1$ is given by $f ^ { ( n ) } ( 1 ) = ( - 1 ) ^ { n } \frac { ( n - 1 ) ! } { 2 ^ { n } }$ for $n \geq 2$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$.
(b) The Taylor series for $f$ about $x = 1$ has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer.
(c) The Taylor series for $f$ about $x = 1$ can be used to represent $f ( 1.2 )$ as an alternating series. Use the first three nonzero terms of the alternating series to approximate $f ( 1.2 )$.
(d) Show that the approximation found in part (c) is within 0.001 of the exact value of $f ( 1.2 )$.

We consider the sequence $\left( u _ { n } \right)$ defined, for all non-zero natural integers $n$, by:
$$u _ { n } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$
The sequence $( v _ { n } )$ is defined by: $v _ { 1 } = u _ { 1 }$, $v _ { 2 } = u _ { 1 } \times u _ { 2 }$ and for all natural integers $n \geqslant 3$, $v _ { n } = u _ { 1 } \times u _ { 2 } \times \ldots \times u _ { n } = v _ { n - 1 } \times u _ { n }$.

1. Verify that we have $v _ { 2 } = \frac { 2 } { 3 }$ then calculate $v _ { 3 }$.
2. We consider the incomplete algorithm below. Copy and complete this algorithm on your paper so that, after its execution, the variable $V$ contains the value $v _ { n }$ where $n$ is a non-zero natural integer defined by the user. No justification is required.
   

   	Algorithm
   1.	$V \leftarrow 1$
   2.	For $i$ varying from 1 to $n$
   3.	$U \leftarrow \frac { \ldots ( \ldots + 2 ) } { ( \ldots + 1 ) ^ { 2 } }$
   4.	$V \leftarrow \ldots$
   5.	End For

   
   
3. a. Show that, for all non-zero natural integers $n$, $u _ { n } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$. b. Show that, for all non-zero natural integers $n$, $0 < u _ { n } < 1$.
4. a. Show that the sequence $( v _ { n } )$ is decreasing. b. Justify that the sequence $( v _ { n } )$ is convergent (we do not ask to calculate its limit).
5. a. Verify that, for all non-zero natural integers $n$, $v _ { n + 1 } = v _ { n } \times \frac { ( n + 1 ) ( n + 3 ) } { ( n + 2 ) ^ { 2 } }$. b. Show by induction that, for all non-zero natural integers $n$, $v _ { n } = \frac { n + 2 } { 2 ( n + 1 ) }$. c. Determine the limit of the sequence $\left( v _ { n } \right)$.
6. We consider the sequence $w _ { n }$ defined by $w _ { 1 } = \ln \left( u _ { 1 } \right)$, $w _ { 2 } = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right)$ and, for all natural integers $n \geqslant 3$, by $$w _ { n } = \sum _ { k = 1 } ^ { n } \ln \left( u _ { k } \right) = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right) + \ldots + \ln \left( u _ { n } \right)$$ Show that $w _ { 7 } = 2 w _ { 1 }$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions of this exercise are independent.

1. We consider the script written in Python language below. \begin{verbatim} def seuil(S) : n=0 u=7 while u < S : n=n+1 u=1.05*u+3 return(n) \end{verbatim} Statement 1: the instruction seuil(100) returns the value 18.
   
2. Let $(S_n)$ be the sequence defined for every natural integer $n$ by $$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$ Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
   
3. Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
   
4. We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$. Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
   
5. Statement 5: $$\int_0^1 x\mathrm{e}^{-x}\,\mathrm{d}x = \frac{\mathrm{e} - 2}{\mathrm{e}}.$$

The distance between the points $A = (1, 2)$ and $B = (4, 6)$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

The value of $\displaystyle\sum_{k=1}^{5} k^2$ is:
(A) 45
(B) 50
(C) 55
(D) 60
(E) 65

The series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ where $a _ { n } = ( - 1 ) ^ { n + 1 } n ^ { 4 } e ^ { - n ^ { 2 } }$
(a) has unbounded partial sums;
(b) is absolutely convergent;
(c) is convergent but not absolutely convergent;
(d) is not convergent, but partial sums oscillate between $-1$ and $+1$.

Let $f$ be continuously differentiable on $\mathbb { R }$. Let $f _ { n } ( x ) = n \left( f \left( x + \frac { 1 } { n } \right) - f ( x ) \right)$. Then,
(a) $f _ { n }$ converges uniformly on $\mathbb { R }$;
(b) $f _ { n }$ converges on $\mathbb { R }$, but not necessarily uniformly;
(c) $f _ { n }$ converges to the derivative of $f$ uniformly on $[ 0,1 ]$;
(d) there is no guarantee that $f _ { n }$ converges on any open interval.

Define $f _ { k } ( n )$ to be the sum of all possible products of $k$ distinct integers chosen from the set $\{ 1,2 , \ldots , n \}$, i.e., $$f _ { k } ( n ) = \sum _ { 1 \leq i _ { 1 } < i _ { 2 } < \ldots < i _ { k } \leq n } i _ { 1 } i _ { 2 } \ldots i _ { k }$$ a) For $k > 1$, write a recursive formula for the function $f _ { k }$, i.e., a formula for $f _ { k } ( n )$ in terms of $f _ { \ell } ( m )$, where $\ell < k$ or ($\ell = k$ and $m < n$). b) Show that $f _ { k } ( n )$, as a function of $n$, is a polynomial of degree $2k$. c) Express $f _ { 2 } ( n )$ as a polynomial in variable $n$.

Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.

Which of the following is/are true for a series of real numbers $\sum a_{n}$?
(A) If $\sum a_{n}$ converges then $\sum a_{n}^{2}$ converges;
(B) If $\sum a_{n}^{2}$ converges then $\sum a_{n}$ converges;
(C) if $\sum a_{n}^{2}$ converges then $\sum \frac{1}{n} a_{n}$ converges;
(D) If $\sum |a_{n}|$ converges then $\sum \frac{1}{n} a_{n}$ converges;

Consider the improper integral $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \log x ) ^ { 2 } } d x$ and the infinite series $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \log k ) ^ { 2 } }$. Which of the following is/are true?
(A) The integral converges but the series does not converge.
(B) The integral does not converge but the series converges.
(C) Both the integral and the series converge.
(D) The integral and the series both fail to converge.

Consider the function $S ( a )$ defined by the limit below: $$S ( a ) : = \lim _ { n \rightarrow \infty } \frac { 1 ^ { a } + 2 ^ { a } + 3 ^ { a } + \cdots + n ^ { a } } { ( n + 1 ) ^ { a - 1 } [ ( n a + 1 ) + ( n a + 2 ) + \cdots + ( n a + n ) ] }$$ Find the sum of all values $a$ such that $S ( a ) = \frac { 1 } { 60 }$.

For a sequence $a _ { i }$ of real numbers, we say that $\sum a _ { i }$ converges if $\lim _ { n \rightarrow \infty } \left( \sum _ { i = 1 } ^ { n } a _ { i } \right)$ is finite. In this question all $a _ { i } > 0$.
Statements
(21) If $\sum a _ { i }$ converges, then $a _ { i } \rightarrow 0$ as $i \rightarrow \infty$. (22) If $a _ { i } < \frac { 1 } { i }$ for all $i$, then $\sum a _ { i }$ converges. (23) If $\sum a _ { i }$ converges, then $\sum ( - 1 ) ^ { i } a _ { i }$ also converges. (24) If $\sum a _ { i }$ does not converge, then $\sum i \tan \left( a _ { i } \right)$ cannot converge.

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.

Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true?
(A) There exists a natural number $M$ such that
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$
(B)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$
(C)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$
(D) For all positive real numbers $R$,
$$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$

Let $\mathbb { R } ^ { + } = \{ x \in \mathbb { R } : x \geq 0 \}$. For $x \in \mathbb { R } ^ { + }$, denote by $\operatorname{FRAC}( x )$ the fractional part of $x$, i.e., $x - n$ where $n$ is the largest integer that is less than or equal to $x$. Consider the series $\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { FRAC } ( x / n ) } { n }$. Pick the correct statement(s) from below.
(A) The above series converges for all $x \in \mathbb { R } ^ { + } - \mathbb { Z }$.
(B) The above series diverges for some non-negative integer $x$.
(C) The above series defines a continuous function in a neighbourhood of $\frac { 1 } { 2 }$.
(D) The above series defines a continuous function in a neighbourhood of 1.

Let $f _ { n } ( x ) = \frac { 1 } { 1 + x ^ { n } }$. Pick the correct statement(s) from below.
(A) $f _ { n }$ converges uniformly on $[ 0,1 / 2 ]$.
(B) $f _ { n }$ converges uniformly on $[ 0,1 )$.
(C) $f _ { n }$ converges uniformly on $[ 0,2 ]$.
(D) $f _ { n }$ converges pointwise on $[ 0 , \infty )$.