##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('5')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;
$a = random(1,9,1);

BEGIN_TEXT

Suppose that \[ \frac{$a}{(1 - x^3)}  =
 \sum_{n=0}^\infty c_n x^n \] $BR
Find the following coefficients of the power series. $BR
\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_1 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_3 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the power series. $BR
\( R = \)  \{ans_rule(20)\} $BR

END_TEXT

$ans0 =  $a ;
$ans1 =  0 ;
$ans2 =  0 ;
$ans3 =  $a ;
$ans4 =  0 ;
$ans5 =  1 ;

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

#DESCRIPTION
#Representation of function as a power series
#ENDDESCRIPTION

#KEYWORDS('Power Series'  )
## tsch tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('14')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);

BEGIN_TEXT

The function 
	\( f(x) = \ln(1 - x^2) \)
is represented as a power series $BR 
	\( f(x) = \sum_{n=0}^\infty c_n x^n .\) $BR
Find the FOLLOWING coefficients in the power series. $BR

\( c_0 = \) \{ans_rule(20)\}  $BR
\( c_1 = \) \{ans_rule(20)\}  $BR
\( c_2 = \) \{ans_rule(20)\}  $BR
\( c_3 = \) \{ans_rule(20)\}  $BR
\( c_4 = \) \{ans_rule(20)\}  $BR

Find the radius of convergence \( R \) of the series. $BR 
\( R = \)  \{ans_rule(20)\}  . $BR

END_TEXT

#@ans=();
$ans0 =  0;
$ans1 =  0;
$ans2 =  -1;
$ans3 =  0;
$ans4 =  -1/2;
$ans5 =  1;

#ANS(num_cmp(relTol=>@ans));
#ANS(ordered_num_cmp_list(@ans) );
ANS(num_cmp($ans0));
ANS(num_cmp($ans1));
ANS(num_cmp($ans2));
ANS(num_cmp($ans3));
ANS(num_cmp($ans4));
ANS(num_cmp($ans5));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

#DESCRIPTION
#Representation of function as a power series
#ENDDESCRIPTION

#KEYWORDS('Power Series'  )
## tsch tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('14')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);

BEGIN_TEXT

The function 
	\( f(x) = $a x \ln(1 + 2x) \)
is represented as a power series $BR 
	\( \displaystyle f(x) = \sum_{n=0}^\infty c_n x^n .\) $BR
Find the FOLLOWING coefficients in the power series. $BR

\( c_0 = \) \{ans_rule(20)\}  $BR
\( c_1 = \) \{ans_rule(20)\}  $BR
\( c_2 = \) \{ans_rule(20)\}  $BR
\( c_3 = \) \{ans_rule(20)\}  $BR
\( c_4 = \) \{ans_rule(20)\}  $BR

Find the radius of convergence \( R \) of the series. $BR 
\( R = \)  \{ans_rule(20)\}  . $BR

END_TEXT

#@ans=();
$ans0 =  0;
$ans1 =  0;
$ans2 =  $a*2;
$ans3 =  $a*(-2);
$ans4 =  $a*8/3;
$ans5 =  1/2;

#ANS(num_cmp(relTol=>@ans));
#ANS(ordered_num_cmp_list(@ans) );
ANS(num_cmp($ans0));
ANS(num_cmp($ans1));
ANS(num_cmp($ans2));
ANS(num_cmp($ans3));
ANS(num_cmp($ans4));
ANS(num_cmp($ans5));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find first few coefficients of power series and find radius of convergence
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('10')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);
$c = random(1,7,1);
$b = $a + $c;
$b1 = -1/($b);

BEGIN_TEXT

Suppose that 
\[ \frac{$a x}{x + $b}  =  \sum_{n=0}^\infty c_n x^n . \] 

Find the following coefficients. $BR
\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_1 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_3 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the power series. $BR
\( R = \)  \{ans_rule(20)\}  $BR

END_TEXT

$ans0 =  0 ;
$ans1 =  "-$a * $b1" ;
$ans2 =  "$ans1 * $b1" ;
$ans3 =  "$ans2 * $b1" ;
$ans4 =  "$ans3 * $b1" ;
$ans5 =  $b ;

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('13')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);
$b = random(2,10,1);

$b1 = 1/$b;

BEGIN_TEXT

The function \(f(x) = \frac{$a}{(1 - $b x)^2} \) is represented as a power series: $BR$BR 
\[ f(x) = \sum_{n=0}^\infty c_n x^n\] $BR
Find the first few coefficients in the power series. $BR
\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_1 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_3 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the series. $BR 
\( R = \)  \{ans_rule(20)\}  . $BR

END_TEXT


$ans0 =  $a ;
$ans1 =  "$a*2*$b" ;
$ans2 =  "$a*3*($b^2)";
$ans3 =  "$a*4*($b^3)";
$ans4 =  "$a*5*($b^4)";
$ans5 =  $b1;

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('17')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;
$a = random(1,10,1);

BEGIN_TEXT

The function 
	\( f(x) = \ln($a - x) \)
is represented as a power series: $BR 
	\[ f(x) = \sum_{n=0}^\infty c_n x^n \] $BR
Find the first few coefficients in the power series. $BR

\( c_0 = \) \{ans_rule(20)\}  $BR
\( c_1 = \) \{ans_rule(20)\}  $BR
\( c_2 = \) \{ans_rule(20)\}  $BR
\( c_3 = \) \{ans_rule(20)\}  $BR
\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the series. $BR 
\( R = \)  \{ans_rule(20)\} $BR

END_TEXT

#@ans=();
$ans0 =  "ln($a)" ;
$ans1 =  "-1/$a" ;
$ans2 =  "-1/(2*$a^2)";
$ans3 =  "-1/(3*$a^3)";
$ans4 =  "-1/(4*$a^4)";
$ans5 =  $a;

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('20')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;
$a = random(2,10,1);
$b = random(2,10,1);

BEGIN_TEXT

The function 
	\( f(x) = $a x \arctan ($b x) \)
is represented as a power series: 
	\[f(x) = \sum_{n=0}^\infty c_n x^n \]
Find the first few coefficients in the power series. $BR

\( c_0 = \) \{ans_rule(20)\}  $BR
\( c_1 = \) \{ans_rule(20)\}  $BR
\( c_2 = \) \{ans_rule(20)\}  $BR
\( c_3 = \) \{ans_rule(20)\}  $BR
\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the series. $BR 

\( R = \)  \{ans_rule(20)\}  . $BR

END_TEXT

$ans0 =  0 ;
$ans1 =  0 ;
$ans2 =  "$a*$b";
$ans3 =  0 ;
$ans4 =  "-1*$a*$b^3/3";
$ans5 =  "1/$b";

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('3')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);
$c = random(1,7,1);
$b = $a + $c;
$b1 = -1/($b);

BEGIN_TEXT

Suppose that \[\frac{$a}{($b + x)}  =
 \sum_{n=0}^\infty c_n x^n \] $BR
Find the following coefficients of the power series. $BR
\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_1 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_3 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence \( R \) of the power series. $BR
\( R = \)  \{ans_rule(20)\}  . $BR

END_TEXT

$ans0 =  "-$a * $b1" ;
$ans1 =  "$ans0 * $b1" ;
$ans2 =  "$ans1 * $b1" ;
$ans3 =  "$ans2 * $b1" ;
$ans4 =  "$ans3 * $b1" ;
$ans5 =  $b ;


ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Power Series', 'Representation of a function as a power series')
##DESCRIPTION
## Find the first few coefficients of a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('9')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;
$a = random(2,10,1);
$b = random(2,10,1);


BEGIN_TEXT

Represent the function \( f(x)= \frac{$a}{(1 - $b x)} \) as a power series: $BR \[ f(x) = \sum_{n=0}^\infty c_n x^n \] $BR

Find the following coefficients: $BR
\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_1 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_3 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR$BR

Find the radius of convergence $BR \( R = \) \{ans_rule(20)\} $BR

END_TEXT

$ans0 =  $a;
$ans1 =  "$a * $b";
$ans2 =  "$a * $b^2";
$ans3 =  "$a * $b^3";
$ans4 =  "$a * $b^4";
$ans5 =  "1/$b";

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );
ANS(num_cmp($ans5) );

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

#DESCRIPTION
#Representation of function as a power series
#ENDDESCRIPTION

#KEYWORDS('Power Series'  )
## tsch tagged and PAID on 3-22-2004

## DBchapter('Infinite Sequences and Series')
## DBsection('Representations of Functions as Power Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.9')
## Problem1('26')


DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(&beginproblem);
$showPartialCorrectAnswers = 0;

$b1 = random(2,6,1) ;

$a = 5 ;
$b  = $b1*8 ;

$c = random(6,12,1);



BEGIN_TEXT

(a)

Evaluate the integral $BR \( \displaystyle \int_{0}^{2} \frac{$b}{x^2+4} dx \). $BR

Your answer should be in the form \( k\pi \), where \( k \) is an integer. What is the value of \( k \)? $BR

Hint: \( \frac{d \arctan(x)}{dx} = \frac{1}{x^2+1} \) $BR

	\( k = \) \{ans_rule(20)\}  $BR 





(b)

Now, lets evaluate the same integral using power series. First, find the power series for the function	\( f(x) = \frac{$b}{x^2+4} \). Then, integrate it from 0 to 2, and call it S. S should be an infinite series. $BR
What are the first few terms of S ? $BR

	\( a_0 = \) \{ans_rule(20)\}  $BR

	\( a_1 = \) \{ans_rule(20)\}  $BR
	\( a_2 = \) \{ans_rule(20)\}  $BR
	\( a_3 = \) \{ans_rule(20)\}  $BR
	\( a_4 = \) \{ans_rule(20)\}  $BR

(c)
The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by \( k \) (the answer to (a)), you have found an estimate for the value of \( \pi \) in terms of an infinite series. Approximate the value of \( \pi \) by the first $a terms. $BR

	\{ans_rule(20)\}  . $BR

END_TEXT

#@ans=();
$ans0 =  $b1;
$ans1 =  $b/2;
$ans2 =  -1*$b/6;
$ans3 =  $b/10;
$ans4 =  -1*$b/14;
$ans5 =  $b/18;
$ans6 =  4*(1-1/3+1/5-1/7+1/9);

#&ANS(std_num_cmp_list(@ans) );
#&ANS(ordered_num_cmp_list(@ans) );
&ANS(std_num_cmp($ans0) );
&ANS(std_num_cmp($ans1) );
&ANS(std_num_cmp($ans2) );
&ANS(std_num_cmp($ans3) );
&ANS(std_num_cmp($ans4) );
&ANS(std_num_cmp($ans5) );
&ANS(std_num_cmp($ans6) );

ENDDOCUMENT();        # This should be the last executable line in the problem.