# DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Absolute Convergence and the Ratio and Root Tests')
# KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test')
# TitleText1('Calculus: Early Transcendentals')
# EditionText1('2')
# AuthorText1('Rogawski')
# Section1('10.5')
# Problem1('11')
# Author('LA Danielson')
# Institution('The College of Idaho')
DOCUMENT();



#Load Necessary Macros

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


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

$exp = Real(random(30,90,10));

($series, $num1, $den1, $num2, $den2, $result, $L, $rho, $compare, $answer, $trueanswer,) = @{list_random(
	[ "\frac{e^n}{n!}", "e^{n+1}", "(n+1)!", "n!", "e^n", "\frac{e}{n+1}", 0, Real(0), "<", "converges", 'convergent'],
	[ "\frac{e^n}{n^n} ", "e^{n+1}", "(n+1)^{n+1}", "n^n", ,"e^n","\frac{e}{n+1}\left( \frac{n}{n+1}\right)^n = \frac{e}{n+1}\left( 1+\frac{1}{n}\right)^{-n}", "0\cdot \frac{1}{e}=0", Real(0), "<", "converges", 'convergent'],
        ["\frac{n^{$exp}}{n!}", "(n+1)^{$exp}", "(n+1)!", "n!", "n^{$exp}", "\frac{1}{n+1}\left( \frac{n+1}{n}\right)^{$exp}=\frac{1}{n+1}\left( 1+\frac{1}{n}\right)^{$exp}", "0\cdot 1=0",  Real(0), "<", "converges", 'convergent' ])};
 




$question = new_multiple_choice();
$question->qa(' \( \sum_{n=1}^{\infty} $series \) is:', 'convergent');
$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');


Context()->texStrings;

BEGIN_TEXT
\{ beginproblem() \}
\{ textbook_ref_exact("Rogawski ET 2e", "10.5", "11") \}
$PAR
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
\[  \sum\limits_{n=1}^{\infty} $series \]
\(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
$PAR

\{ $question->print_q() \}
\{ $question->print_a() \}
END_TEXT

Context()->normalStrings;


ANS($rho->cmp);
ANS(radio_cmp($question->correct_ans));

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
With \( a_n = $series \), 
\[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{$num1}{$den1} \cdot \frac{$num2}{$den2} = $result \] and \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = $L $compare 1. \]
Therefore, the series \( \sum\limits_{n=1}^{\infty} $series \) $answer by the Ratio Test.


END_SOLUTION

ENDDOCUMENT()