View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.53.pg

    1 # DBsubject('Calculus')
    2 # DBchapter('')
    3 # DBsection('')
    4 # KEYWORDS('')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.5')
    9 # Problem1('53')
   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 #sin (1/n^2)
   27 
   28 $exp = random(2, 9);
   29 
   30 $denominator = "n^{$exp}";
   31 
   32 $sol1 = "Because \[\lim_{n\to\infty} \cos\left(\frac{1}{$denominator}\right) = \cos 0 = 1 \ne 0,\] the general term in the series \( \sum\limits_{n=1}^{\infty} (-1)^n \cos\left(\frac{1}{$denominator}\right)\) does not tend toward zero; therefore, the series diverges by the Divergence Test.";
   33 
   34 $sol2 = "Here, we will apply the Limit Comparison Test, comparing with the \(p\)-series with \( p = $exp \).  Now,
   35 \[ L = \lim_{n \to \infty} \frac{\sin \left(\frac{1}{$denominator}\right)}{\frac{1}{n^{$exp}}} = \lim_{u \to 0} \frac{\sin u}{u} = 1, \]
   36 where \(u = \frac{1}{n^{$exp}}\).  The \(p\)-series with \(p = $exp\) converges and \(L\) exists; therefore, the series \( \sum\limits_{n=1}^{\infty} \sin \left(\frac{1}{$denominator}\right) \) also converges.";
   37 
   38 ($an, $solution, $trueanswer) = @{list_random(
   39   [ "(-1)^n \cos \left(\frac{1}{$denominator}\right)", $sol1, 'divergent'],
   40   [ "\sin \left(\frac{1}{$denominator}\right)", $sol2, 'convergent'] )};
   41 
   42 
   43 #Let's try to make a multiple choice question
   44 $question = new_multiple_choice();
   45 $question->qa(' \( \sum\limits_{n=1}^{\infty} $an \) is:', $trueanswer);
   46 $question->makeLast( 'convergent', 'divergent');
   47 
   48 
   49 Context()->texStrings;
   50 
   51 BEGIN_TEXT
   52 \{ beginproblem() \}
   53 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "53") \}
   54 $PAR
   55 Determine convergence or divergence using any method covered so far.
   56 $PAR
   57 \{ $question->print_q() \}
   58 \{ $question->print_a() \}
   59 END_TEXT
   60 
   61 Context()->normalStrings;
   62 
   63 #Answer Check Time!
   64 ANS(radio_cmp($question->correct_ans));
   65 
   66 Context()->texStrings;
   67 SOLUTION(EV3(<<'END_SOLUTION'));
   68 $PAR
   69 $SOL
   70 $solution
   71 
   72 END_SOLUTION
   73 
   74 ENDDOCUMENT()