# DBsubject('Calculus')
# DBchapter('Infinite Series and Sequences')
# DBsection('Absolute Convergence and the Root and Ratio Tests')
# KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test')
# TitleText1('Calculus: Early Transcendentals')
# EditionText1('2')
# AuthorText1('Rogawski')
# Section1('10.5')
# Problem1('5')
# Author('Emily Price')
# Institution('W.H.Freeman')
DOCUMENT();



#Load Necessary Macros

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


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

#Book Values
# $exp1 = 2
# $exp2 = 1
# $constant = 1

$exp1 = random(2, 7);
$exp2 = random(1,$exp1-1,1);#Formula("$exp1 - 1")->reduce;
$constant = random(1, 9);
$num = Formula("n^{$exp2}")->reduce;
$num2 = Formula("(n+1)^{$exp2}")->reduce;

$denominator = "n^{$exp1} + $constant";
$denomplus1 = "(n+1)^{$exp1} + $constant";

$rho = Real(1);

#Let's try to make a multiple choice question
$question = new_multiple_choice();
$question->qa(' \( \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \) is:', 'The Ratio Test is inconclusive');
$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');


Context()->texStrings;

BEGIN_TEXT
\{ beginproblem() \}
\{ textbook_ref_exact("Rogawski ET 2e", "10.5", "5") \}
$PAR
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
\[  \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \]
\(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
$PAR
\{ $question->print_q() \}
\{ $question->print_a() \}
END_TEXT

Context()->normalStrings;

#Answer Check Time!
ANS($rho->cmp);
ANS(radio_cmp($question->correct_ans));

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
With \( a_n = \frac{$num}{$denominator} \),
 
\[ \left| \frac{a_{n+1}}{a_n} \right| = 
\frac{$num2}{$denomplus1} \cdot \frac{$denominator}{$num} = \frac{$num2}{$num} \cdot \frac{$denominator}{$denomplus1}, \]
and
\[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \cdot 1 = 1. \]
Therefore, for the series \( \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \), the Ratio Test is inconclusive.

END_SOLUTION

ENDDOCUMENT()