View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.17.pg

    1 # DBsubject('Calculus')
    2 # DBchapter('Infinite Series and Sequences')
    3 # DBsection('Absolute Convergence and the Root and Ratio Tests')
    4 # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.5')
    9 # Problem1('17')
   10 # Author('LA Danielson')
   11 # Institution('The College of Idaho')
   12 DOCUMENT();
   13 
   14 
   15 
   16 #Load Necessary Macros
   17 
   18 loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", );
   19 loadMacros("Parser.pl");
   20 loadMacros("freemanMacros.pl");
   21 
   22 
   23 Context()->variables->add(n=>'Real');
   24 
   25 #Book Values
   26 #numerator = n^2
   27 #denominator = (2n + 1)!
   28 
   29 $a = random(2,9,1);
   30 
   31 
   32 ($series, $num1, $den1, $num2, $den2, $result, $L, $rho, $compare, $answer, $trueanswer,) = @{list_random(
   33   [ "\frac{(n!)^2}{(2n)!}", "((n+1)!)^2", "(2n+2)!", "(2n)!", "(n!)^2", "\frac{(n+1)^2}{(2n+2)(2n+1)}", "\frac{1}{4}", Real(1/4), "<", "converges", 'convergent'],
   34   [ "\frac{$a n^2}{(2n+1)!}", "$a (n+1)^2",  "(2n+3)!",  "(2n+1)!", "$a n^2", "\left(\frac{n+1}{n} \right)^2 \frac{1}{(2n+3)(2n+2)}", "1^2\cdot 0 = 0", Real(0), "<", "converges", 'convergent'] )};
   35 
   36 
   37 
   38 
   39 #Let's try to make a multiple choice question
   40 $question = new_multiple_choice();
   41 $question->qa(' \( \sum_{n=1}^{\infty} $series \) is:', 'convergent');
   42 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   43 
   44 
   45 Context()->texStrings;
   46 
   47 BEGIN_TEXT
   48 \{ beginproblem() \}
   49 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "17") \}
   50 $PAR
   51 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
   52 \[  \sum\limits_{n=1}^{\infty} $series \]
   53 \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   54 $PAR
   55 
   56 \{ $question->print_q() \}
   57 \{ $question->print_a() \}
   58 END_TEXT
   59 
   60 Context()->normalStrings;
   61 
   62 ANS($rho->cmp);
   63 ANS(radio_cmp($question->correct_ans));
   64 
   65 Context()->texStrings;
   66 SOLUTION(EV3(<<'END_SOLUTION'));
   67 $PAR
   68 $SOL
   69 
   70 With \( a_n = $series \),
   71 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{$num1}{$den1} \cdot \frac{$num2}{$den2} = $result \] and \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = $L $compare 1. \]
   72 Therefore, the series \( \sum\limits_{n=1}^{\infty} $series \) $answer by the Ratio Test.
   73 
   74 
   75 
   76 END_SOLUTION
   77 
   78 ENDDOCUMENT()