View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.5.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('5')
   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 # $exp1 = 2
   27 # $exp2 = 1
   28 # $constant = 1
   29 
   30 $exp1 = random(2, 7);
   31 $exp2 = random(1,$exp1-1,1);#Formula("$exp1 - 1")->reduce;
   32 $constant = random(1, 9);
   33 $num = Formula("n^{$exp2}")->reduce;
   34 $num2 = Formula("(n+1)^{$exp2}")->reduce;
   35 
   36 $denominator = "n^{$exp1} + $constant";
   37 $denomplus1 = "(n+1)^{$exp1} + $constant";
   38 
   39 $rho = Real(1);
   40 
   41 #Let's try to make a multiple choice question
   42 $question = new_multiple_choice();
   43 $question->qa(' \( \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \) is:', 'The Ratio Test is inconclusive');
   44 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   45 
   46 
   47 Context()->texStrings;
   48 
   49 BEGIN_TEXT
   50 \{ beginproblem() \}
   51 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "5") \}
   52 $PAR
   53 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.
   54 \[  \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \]
   55 \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   56 $PAR
   57 \{ $question->print_q() \}
   58 \{ $question->print_a() \}
   59 END_TEXT
   60 
   61 Context()->normalStrings;
   62 
   63 #Answer Check Time!
   64 ANS($rho->cmp);
   65 ANS(radio_cmp($question->correct_ans));
   66 
   67 Context()->texStrings;
   68 SOLUTION(EV3(<<'END_SOLUTION'));
   69 $PAR
   70 $SOL
   71 With \( a_n = \frac{$num}{$denominator} \),
   72 
   73 \[ \left| \frac{a_{n+1}}{a_n} \right| =
   74 \frac{$num2}{$denomplus1} \cdot \frac{$denominator}{$num} = \frac{$num2}{$num} \cdot \frac{$denominator}{$denomplus1}, \]
   75 and
   76 \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \cdot 1 = 1. \]
   77 Therefore, for the series \( \sum\limits_{n=1}^{\infty} \frac{$num}{$denominator} \), the Ratio Test is inconclusive.
   78 
   79 END_SOLUTION
   80 
   81 ENDDOCUMENT()