View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.31.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('31')
   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 #$rho = 1/3, 3^n a_n
   27 
   28 $den = random(2,9,1);
   29 $base = random(2,9,1);
   30 $rho = Real($base/$den);
   31 $display = "\frac{$base}{$den}";
   32 
   33 $bn = "$base^{n} a_n";
   34 
   35 if($den==$base){
   36    $trueanswer = 'The Ratio Test is inconclusive';
   37    $compare = "=";
   38    $FinalState = "the Ratio Test is inconclusive for the series \( \sum\limits_{n=1}^{\infty} $bn .\)";
   39 }elsif($den>$base){
   40    $trueanswer = 'convergent';
   41    $compare = "<";
   42    $FinalState = "the series \( \sum\limits_{n=1}^{\infty} $bn \) converges by the Ratio Test.";
   43 }else{
   44    $trueanswer = 'divergent';
   45    $compare = ">";
   46    $FinalState = "the series \( \sum\limits_{n=1}^{\infty} $bn \) diverges by the Ratio Test.";
   47 }
   48 
   49 $question = new_multiple_choice();
   50 $question->qa(' \( \sum\limits_{n=1}^{\infty} $bn \) is:', $trueanswer);
   51 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   52 
   53 
   54 Context()->texStrings;
   55 
   56 BEGIN_TEXT
   57 \{ beginproblem() \}
   58 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "31") \}
   59 $PAR
   60 Assume that \( | \frac{a_{n+1}}{a_n}|\) converges to \( \rho = \frac{1}{$den} \).  What can you say about the convergence of the given series?
   61 \[  \sum\limits_{n=1}^{\infty} {b_n} =  \sum\limits_{n=1}^{\infty} $bn\]
   62 \(\lim\limits_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   63 $PAR
   64 \{ $question->print_q() \}
   65 \{ $question->print_a() \}
   66 END_TEXT
   67 
   68 Context()->normalStrings;
   69 
   70 #Answer Check Time!
   71 ANS($rho->cmp);
   72 ANS(radio_cmp($question->correct_ans));
   73 
   74 Context()->texStrings;
   75 SOLUTION(EV3(<<'END_SOLUTION'));
   76 $PAR
   77 $SOL
   78 Let \( b_n = $bn \).  Then
   79 \[ \rho  = \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| = \lim_{n \to \infty} \frac{$base^{n+1}}{$base^n} \left| \frac{a_{n+1}}{a_n} \right|
   80  =  $base \cdot \frac{1}{$den} = $display $compare 1. \]
   81 Therefore, $FinalState
   82 
   83 
   84 END_SOLUTION
   85 
   86 ENDDOCUMENT()