#DESCRIPTION
# Series.  Convergence
#ENDDESCRIPTION

#KEYWORDS('Series' , 'Convergence' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('9,20')

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

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

# No partial credit on this problem, so we say:
install_problem_grader(~~&std_problem_grader);

TEXT(beginproblem());
$showPartialCorrectAnswers = 0;

$a = 2;
$b = random(2,6,2);
$f = 2**(2*$b);
$g = 2**($b/2);

BEGIN_TEXT

Given:$BR
\( \displaystyle A_n = \frac{$f^{n/2}}{$g^{2n}} \) $BR
Determine: $BR
(a) whether \( \displaystyle \sum_{n=1}^\infty( A_n ) \) is convergent. \{ans_rule(10)\}$BR
(b) whether \( \lbrace A_n \rbrace \)is convergent. \{ans_rule(10)\} $BR 
If convergent, enter the limit of convergence. If not, enter "DIV" (unquoted). 

END_TEXT

$ansA = 'DIV';
$ansB = 1 ;

ANS(num_cmp($ansA, strings=>['DIV']));
ANS(num_cmp($ansB, strings=>['DIV']));

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

#DESCRIPTION
# Convert the given decimal into a rational fraction.
#KEYWORDS('Rational Numbers' , 'Conversion', 'Decimal to Fraction' )
#ENDDESCRIPTION
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('3, 9')

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(20,90,10);
$b = $a/10;
$c = $b - 1;

BEGIN_TEXT

Given:$BR
\( \displaystyle A_n = \frac{$a}{$b^{n}} \) $BR
Determine:$BR
(a) whether \( \displaystyle \sum_{n=1}^\infty( A_n ) \) is convergent. \{ans_rule(30)\} $BR
(b) whether \( \lbrace A_n \rbrace \)is convergent. \{ans_rule(30)\} $BR
If convergent, enter the limit of convergence. If not, enter "DIV" (unquoted).

END_TEXT

$ansP = $a/$c ;
ANS(num_cmp($ansP, strings=>['DIV']));
$ansQ = 0 ;
ANS(num_cmp($ansQ, strings=>['DIV']));

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

#DESCRIPTION
# Series.  Convergence
#ENDDESCRIPTION

#KEYWORDS('Series' , 'Convergence' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('9, 20')

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

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

# No partial credit on this problem, so we say:
install_problem_grader(~~&std_problem_grader);

TEXT(beginproblem());
$showPartialCorrectAnswers = 0;

$a = random(20,90,10);
$b = $a/10;

BEGIN_TEXT

Given:$BR
\( \displaystyle A_n = \frac{$b^{n}}{$a} \) $BR
Determine: $BR
(a) whether \( \displaystyle \sum_{n=1}^\infty( A_n ) \) is convergent. \{ans_rule(30)\} $BR
(b) whether \( \lbrace A_n \rbrace \)is convergent. \{ans_rule(30)\} $BR
If convergent, enter the limit of convergence. If not, enter 'DIV'
(unquoted).

END_TEXT

$ansA = 'DIV';
$ansB = 'DIV';

ANS(num_cmp($ansA, strings=>['DIV']));
ANS(num_cmp($ansB, strings=>['DIV']));

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

##KEYWORDS('Series' , 'Sequence', 'Convergence' )
##DESCRIPTION
## Determine if a series converges and evaluate it. Also determine if a 
## sequence is convergent and evaluate it 
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## 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(1,4,1);
$b = random(5,11,1);
$c = random(1,11,1);

BEGIN_TEXT
Given: \( A_n = \frac{$a n}{$b n + $c} \) determine:$BR
(a) whether \( \displaystyle \sum_{n=1}^\infty( A_n ) \) is convergent: \{ans_rule(10)\}$BR$BR
(b) whether \( \lbrace A_n \rbrace \)is convergent: \{ans_rule(10)\}
$BR$BR
$BBOLD Note: $EBOLD If convergent, enter the limit of convergence. If not convergent, enter $BITALIC div $EITALIC.)

END_TEXT

ANS(num_cmp('div', strings=>['div']) );
ANS(num_cmp("$a/$b", strings=>['div']) );

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

#DESCRIPTION
# Find Sum of given Series.
#ENDDESCRIPTION

#KEYWORDS('Series' , 'Summation' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('7')

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= non_zero_random(-3,10,1);

BEGIN_TEXT
For the following series, if it converges, enter the limit of convergence.
If not, enter "DIV" (unquoted). $BR
\[ \sum_{n=1}^\infty \ln(2(n+1)) - \ln(2n) \]
\{ans_rule(20)\}
END_TEXT

$answer = 'DIV';
ANS(num_cmp($answer, strings=>['DIV']));

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

#DESCRIPTION
# Find Sum of given Series.
#ENDDESCRIPTION

#KEYWORDS('Series' , 'Summation' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('7')

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= non_zero_random(-3,10,1);

BEGIN_TEXT
Determine the sum of the following series. $BR
\[ \sum_{n=1}^\infty \left( \sin \left( \frac{$a}{n} \right) -\sin \left( \frac{$a}{n+1}\right)\right) \]
\{ans_rule(20)\}
END_TEXT

$answer = sin($a);
ANS(num_cmp($answer));

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

#DESCRIPTION
# Convert the given decimal into a rational fraction.
#KEYWORDS('Rational Numbers' , 'Conversion', 'Decimal to Fraction' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('37')

#ENDDESCRIPTION

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(2,10,1);
$b = random(6,10,1);
$frac = (11*$a + $b)/(11);
$nicefrac = spf($frac, "%14.12g");

BEGIN_TEXT
Express  \( $nicefrac . . . \) as a rational number, in the form \( \frac{p}{q} \)
$BR where \(p\) and \(q\) are positive integers
with no common factors.$BR
\(p \) = \{ans_rule(30)\} and 
\(q \) = \{ans_rule(30)\}
END_TEXT

$ansP = 11*$a +$b ;
ANS(num_cmp($ansP));
$ansQ = 11 ;
ANS(num_cmp($ansQ));

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

#DESCRIPTION
# Convert the given decimal into a rational fraction.
#KEYWORDS('Rational Numbers' , 'Conversion', 'Decimal to Fraction' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('40')

#ENDDESCRIPTION

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(2,10,1);
$b = random(67,103,9);
$frac = (111*$a + $b)/(111);
$c = ($b/111)/1000000000;
$nicefrac = $frac - $c;

BEGIN_TEXT
Express  \( $nicefrac . . . \) as a rational number, in the form \( \frac{p}{q} \)
$BR where \(p\) and \(q\) have no common factors.$BR
\(p \) = \{ans_rule(30)\} and 
\(q \) = \{ans_rule(30)\}
END_TEXT

$ansP = 111*$a +$b ;
ANS(num_cmp($ansP));
$ansQ = 111 ;
ANS(num_cmp($ansQ));

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

#DESCRIPTION
# Series. Find sum
#ENDDESCRIPTION

#KEYWORDS('Series' , 'Summation' )
## tsuda tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('17')


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;

$c = non_zero_random(-4,4,1);
$d = random(5,9,1);
$a = $c**-1;
$r = $c/$d ;

BEGIN_TEXT
Determine the sum of the following series. $BR
\[ \sum_{n=1}^\infty \frac{($c)^{n-1}}{$d^n} \]
\{ans_rule(20)\}
END_TEXT

$answer = $a/(1-$r) - $a;

ANS(num_cmp($answer));

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

#DESCRIPTION
# Find Sum of given Series.
#ENDDESCRIPTION

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

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('56')


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(10,80,5);
$b = random(10,25,1);
$r = $a/100;
$d = $b + $r*$b;
$f = 1 - $r;

BEGIN_TEXT
A ball drops from a height of $b feet. Each time it hits the ground, it bounces
up $a percents of the height it fall. Assume it goes on forever, find the total
distance it travels.

$BR

\{ans_rule(20)\}
END_TEXT

$answer = $d/$f;
ANS(num_cmp($answer));

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

##KEYWORDS('Series' , 'Divergent', 'Convergent' )
##DESCRIPTION
## Determine if a series is convergent and evaluate it
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('23')

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

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

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

$a = random(1,19,1);

BEGIN_TEXT
Determine whether the series is convergent or divergent. If convergent, find the sum;
if divergent, enter $BITALIC div $EITALIC.
\[ \sum_{n=1}^\infty \frac{$a}{n(n+2)} \]
Answer: \{ans_rule(20)\}
END_TEXT


ANS(num_cmp("$a*3/4", strings=>['div']));

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

##KEYWORDS('Series' , 'Divergent', 'Convergent' )
##DESCRIPTION
## Determine if a series is convergent and evaluate it
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('31')

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

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

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

$a = random(2,19,1);

BEGIN_TEXT
Determine whether the series is convergent or divergent. If convergent, find the sum;
if divergent, enter $BITALIC div $EITALIC. $BR
\[ \sum_{n=1}^\infty \arctan($a n) \]
Answer: \{ans_rule(20)\}
END_TEXT


ANS(num_cmp('div', strings=>['div']) );

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

##KEYWORDS('Series' , 'Divergent', 'Convergent' )
##DESCRIPTION
## Determine if a series is convergent and evaluate it
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('27')

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

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

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

$a = random(1,19,1);

BEGIN_TEXT
Determine whether the series is convergent or divergent. If convergent, find the sum;
if divergent, enter $BITALIC div $EITALIC.
\[ \sum_{n=1}^\infty \frac{n}{\sqrt{n^2+$a}} \]
Answer: \{ans_rule(20)\}
END_TEXT

ANS(num_cmp('div', strings=>['div']) );

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

##KEYWORDS('Series' , 'Geometric Series')
##DESCRIPTION
## Determine conditions that will make the series converge
##ENDDESCRIPTION

## Shotwell cleaned


## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('41')

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

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

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

$a = random(2,9,1);

BEGIN_TEXT
Find the values of \(x\) for which the series below converges.
\[ \sum_{n=1}^\infty \frac{x^n}{$a^n} \]
Answer: \(|x|<\) \{ans_rule(10)\}
END_TEXT

ANS(num_cmp($a));

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

##KEYWORDS('Series' , 'Geometric Series')
##DESCRIPTION
## Determine what will make the series converge
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.2')
## Problem1('43')

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

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

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

$a = random(2,9,1);

BEGIN_TEXT
Find the values of \(x\) for which the series below converges.
\[ \sum_{n=1}^\infty x^n $a^n \]
Answer: \(|x|<\) \{ans_rule(10)\}
END_TEXT

ANS(num_cmp("1/$a") );

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