View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.11.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('11')
   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 $exp = Real(random(30,90,10));
   26 
   27 ($series, $num1, $den1, $num2, $den2, $result, $L, $rho, $compare, $answer, $trueanswer,) = @{list_random(
   28   [ "\frac{e^n}{n!}", "e^{n+1}", "(n+1)!", "n!", "e^n", "\frac{e}{n+1}", 0, Real(0), "<", "converges", 'convergent'],
   29   [ "\frac{e^n}{n^n} ", "e^{n+1}", "(n+1)^{n+1}", "n^n", ,"e^n","\frac{e}{n+1}\left( \frac{n}{n+1}\right)^n = \frac{e}{n+1}\left( 1+\frac{1}{n}\right)^{-n}", "0\cdot \frac{1}{e}=0", Real(0), "<", "converges", 'convergent'],
   30         ["\frac{n^{$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' ])};
   31 
   32 
   33 
   34 
   35 
   36 $question = new_multiple_choice();
   37 $question->qa(' \( \sum_{n=1}^{\infty} $series \) is:', 'convergent');
   38 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   39 
   40 
   41 Context()->texStrings;
   42 
   43 BEGIN_TEXT
   44 \{ beginproblem() \}
   45 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "11") \}
   46 $PAR
   47 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
   48 \[  \sum\limits_{n=1}^{\infty} $series \]
   49 \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   50 $PAR
   51 
   52 \{ $question->print_q() \}
   53 \{ $question->print_a() \}
   54 END_TEXT
   55 
   56 Context()->normalStrings;
   57 
   58 
   59 ANS($rho->cmp);
   60 ANS(radio_cmp($question->correct_ans));
   61 
   62 Context()->texStrings;
   63 SOLUTION(EV3(<<'END_SOLUTION'));
   64 $PAR
   65 $SOL
   66 With \( a_n = $series \),
   67 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{$num1}{$den1} \cdot \frac{$num2}{$den2} = $result \] and \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = $L $compare 1. \]
   68 Therefore, the series \( \sum\limits_{n=1}^{\infty} $series \) $answer by the Ratio Test.
   69 
   70 
   71 END_SOLUTION
   72 
   73 ENDDOCUMENT()