View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.1.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('1')
   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 #numerator = 1;
   27 #denominator = 5^n
   28 
   29 #Random Values
   30 $base1 = Real(random(2, 9));
   31 
   32 
   33 $denominator = "$base1^n";
   34 $rho = 1/$base1;
   35 
   36 ($series, $num, $den1, $den2, $trueanswer,) = @{list_random(
   37   [ "\frac{1}{$denominator}", "1", "1", "$base1", 'convergent'],
   38   [ "\frac{(-1)^{n-1}n}{$denominator} ", "n+1", "n", "$base1 n", 'convergent'])};
   39 
   40 
   41 
   42 
   43 #Let's try to make a multiple choice question
   44 $question = new_multiple_choice();
   45 $question->qa(' \( \sum\limits_{n=1}^{\infty} $series \) is:', $trueanswer);
   46 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   47 
   48 
   49 Context()->texStrings;
   50 
   51 BEGIN_TEXT
   52 \{ beginproblem() \}
   53 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "1") \}
   54 $PAR
   55 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
   56 \[  \sum\limits_{n=1}^{\infty} $series \]
   57 \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   58 $PAR
   59 \{ $question->print_q() \}
   60 \{ $question->print_a() \}
   61 END_TEXT
   62 
   63 Context()->normalStrings;
   64 
   65 #Answer Check Time!
   66 ANS($rho->cmp);
   67 ANS(radio_cmp($question->correct_ans));
   68 
   69 Context()->texStrings;
   70 SOLUTION(EV3(<<'END_SOLUTION'));
   71 $PAR
   72 $SOL
   73 With \( a_n = $series \),
   74 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{$num}{$base1^{n+1}} \cdot \frac{$base1^n}{$den1} = \frac{$num}{$den2} \] and \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{$base1} < 1. \]
   75 Therefore, the series \( \sum\limits_{n=1}^{\infty} $series \) converges by the Ratio Test.
   76 
   77 
   78 END_SOLUTION
   79 
   80 ENDDOCUMENT()