View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.7.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('7')
   10 # Author('Emily Price')
   11 # Institution('W.H.Freeman')
   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 # $exp = 100
   27 # $base = 2
   28 
   29 $exp = random(100, 200);
   30 $base = random(2, 20);
   31 $a = random(3, 9,1); #LAD added for 10.5.8
   32 $am1 = $a-1;
   33 
   34 ($series, $num1, $den1, $num2, $den2, $result, $L, $rho, $compare, $answer, $trueanswer,) = @{list_random(
   35   [ "\frac{$base^n}{n^{$exp}}","$base^{n+1}", "(n+1)^{$exp}", "n^{$exp}", "$base^n", "$base \left(\frac{n}{n+1}\right)^{$exp}", "$base\cdot 1^{$exp} = $base", Real($base), ">", "diverges", 'divergent'],
   36   [ "\frac{n^{$a}}{$a^{n^{$am1}}} ", "(n+1)^{$a}", "$a^{(n+1)^{$am1}}", "$a^{n^{$am1}}", "n^{$a}","\left( \frac{n+1}{n}\right)^{$a}\cdot\frac{1}{$a^{(n+1)^{$am1}-n^{$am1}}} ","1^{$a}\cdot 0=0", Real(0), "<", "converges", 'convergent'])};
   37 
   38 
   39 
   40 #Let's try to make a multiple choice question
   41 $question = new_multiple_choice();
   42 $question->qa(' \( \sum\limits_{n=1}^{\infty}$series  \) is:', $trueanswer);
   43 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   44 
   45 
   46 Context()->texStrings;
   47 
   48 BEGIN_TEXT
   49 \{ beginproblem() \}
   50 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "7") \}
   51 $PAR
   52 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
   53 \[  \sum\limits_{n=1}^{\infty} $series \]
   54 \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   55 $PAR
   56 \{ $question->print_q() \}
   57 \{ $question->print_a() \}
   58 END_TEXT
   59 
   60 Context()->normalStrings;
   61 
   62 #Answer Check Time!
   63 ANS($rho->cmp);
   64 ANS(radio_cmp($question->correct_ans));
   65 
   66 Context()->texStrings;
   67 SOLUTION(EV3(<<'END_SOLUTION'));
   68 $PAR
   69 $SOL
   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 
   77 END_SOLUTION
   78 
   79 ENDDOCUMENT()