View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.35.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', 'root test')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.5')
    9 # Problem1('35')
   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 $ch = random(0,1,1);
   26 
   27 if($ch>0){#convergent p
   28    $p = random(2,9,1);
   29    $answer = "converges";
   30 }else{
   31    $p = random(.1,.9,.1);
   32    $answer = "diverges";
   33 }
   34 
   35 
   36 #Let's try to make a multiple choice question
   37 $question = new_multiple_choice();
   38 $question->qa("The \( p \)-series \( \sum\limits_{n=1}^{\infty} \frac{1}{n^{$p}} \) $answer by the Ratio Test.", 'False');
   39 $question->makeLast('True', 'False');
   40 
   41 Context()->texStrings;
   42 
   43 BEGIN_TEXT
   44 \{ beginproblem() \}
   45 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "35") \}
   46 $PAR
   47 Determine if the following statment is True or False:
   48 $PAR
   49 \{ $question->print_q() \}
   50 \{ $question->print_a() \}
   51 END_TEXT
   52 
   53 Context()->normalStrings;
   54 
   55 #Answer Check Time!
   56 ANS(radio_cmp($question->correct_ans));
   57 
   58 Context()->texStrings;
   59 SOLUTION(EV3(<<'END_SOLUTION'));
   60 $PAR
   61 $SOL
   62 With \( a_n = \frac{1}{n^{$p}} \),
   63 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{(n+1)^{$p}} \cdot \frac{n^{$p}}{1} = \left( \frac{n}{n+1} \right)^{$p} \quad \textrm{and} \quad \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1^{$p} = 1. \]
   64 Therefore, the Ratio Test is inconclusive for the \( p \)-series  \( \sum\limits_{n=1}^\infty \frac1{n^{$p}} \).
   65 
   66 
   67 END_SOLUTION
   68 
   69 ENDDOCUMENT()