Series Test

Series: limit comparison test

Complete Code

POD for Macro Files

PG problem file

Explanation

DOCUMENT();

loadMacros(
    'PGstandard.pl', 'PGML.pl',
    'niceTables.pl', 'parserPopUp.pl',
    'PGgraders.pl',  'parserMultiAnswer.pl',
    'PGcourse.pl'
);

Preamble

We load niceTables.pl to create a table in which answer blanks are stacked on top of each other to form a fraction. We use PGgraders.pl to give partial credit incrementally. We use parserMultiAnswer.pl for the fraction answer so that we can accept two correct answers, depending on how much a student has simplified their answer.

Context()->variables->are(n => 'Real');

$a = random(2, 9);
$b = random(2, 9);
$c = random(5, 20);
$d = random(3, 9);
$e = random(2, 9);

$dm1 = $d - 1;
$dm2 = $d - 2;

# TeX
$series   = "\sum_{n=$c}^{\infty} \frac{$a n + $b}{$c n^{$d} + $e}";
$fraction = "\lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty}";

$num1 = Formula("$a n^$d + $b n^$dm1");
$den1 = Formula("$c n^$d + $e");

$num2 = Formula("$a + $b/n");
$den2 = Formula("$c + $e/(n^$d)");

$multians = MultiAnswer($num1, $den1)->with(
    singleResult => 0,
    checker      => sub {
        my ($correct, $student, $ansHash) = @_;
        my ($stu1, $stu2) = @{$student};

        if (($num1 == $stu1 && $den1 == $stu2)
            || ($num2 == $stu1 && $den2 == $stu2))
        {
            return [ 1, 1 ];
        } elsif (($num1 == $stu1 && $den2 == $stu2)
            || ($num2 == $stu1 && $den1 == $stu2))
        {
            $ansHash->setMessage(1, "Check your algebra");
            $ansHash->setMessage(2, "Check your algebra");
            return [ 0, 0 ];
        } elsif ($num1 == $stu1 || $num2 == $stu1) {
            return [ 1, 0 ];
        } elsif ($den1 == $stu2 || $den2 == $stu2) {
            return [ 0, 1 ];
        } else {
            return [ 0, 0 ];
        }
    }
);

$limit = Formula("$a/$c");
$popup =
    PopUp([ 'Choose', 'Converges', 'Diverges', 'Inconclusive' ], 'Converges');

# Display the fraction and answer blanks nicely
$frac = LayoutTable(
    [ [ [ ans_rule(10), rowbottom => 1 ] ], [ ans_rule(10) ] ],
    center     => 0,
    allcellcss => { padding => '4pt' }
);

Setup

We use the MultiAnswer object $multians to allow students to enter one of two correct answers. We could have also accomplished this using two custom answer checkers.

We display the answerblanks nicely as a fraction in HTML and TeX modes by how we constructed $showfraction.

BEGIN_PGML
Use the limit comparison test to determine whether
[``\sum_{n=[$c]}^{\infty} a_n = \sum_{n=[$c]}^{\infty} \frac{[$a] n + [$b]}{[$c] n^{[$d]} + [$e]}``]
converges or diverges.

a. Choose a series [``\sum_{n=[$c]}^\infty b_n``] with terms of the form
[``b_n = \frac{1}{n^p}``] and apply the limit comparison test.  Write your
answer as a fully reduced fraction.  For [``n \geq [$c]``],
[```\frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n}```][$frac]* [@ ANS($multians->cmp); '' @]

b. Evaluate the limit in the previous part.  Enter [` \infty `] as _infinity_
and [` -\infty `] as _-infinity_.  If the limit does not exist, enter _DNE_.

[``\lim_{n\to\infty} \frac{a_{n}}{b_{n}} =``] [_]{$limit}{15}

c. By the limit comparison test, does the series converge, diverge, or is the
test inconclusive?  [_]{$popup}
END_PGML

Statement

Most of this is standard latex markup in a PGML block. Note that to display the fraction above, we use [$frac]* followed by the PGML codeblock [@ ANS($multians->cmp); '' @] which does the answer checking using the multianswer described above. There is a '' at the tend of the codeblock to return an empty string instead of a HASHREF which we get from the ANS method.

install_problem_grader(~~&custom_problem_grader_fluid);

$ENV{grader_numright} = [ 2,   4 ];
$ENV{grader_scores}   = [ 0.4, 1 ];
$ENV{grader_message} =
    'You can earn 40% partial credit for 2 - 3 correct answers.';

Answer

We use the problem grader fluid to give partial credit incrementally: 0% for 0-1 correct answers, 40% for 2-3 correct answers, and full credit for 4 correct answers.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.