View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.39.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('39')
   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 = n
   27 #denominator = 3n + 1
   28 
   29 ##making the top and bottom in the form an + b
   30 
   31 $btop = random(1, 19,2);
   32 $abottom = random(2, 9);
   33 $bbottom = random(2, 20,2);
   34 
   35 $numerator = "n + $btop";
   36 $denominator = "$abottom n + $bbottom";
   37 
   38 ($an, $num, $den, $result, $L, $FinalState, $trueanswer,) = @{list_random(
   39   [ "\left(\frac{$numerator}{$denominator}\right)^n", "$numerator", "$denominator", "\frac{1}{$abottom} < 1", Real(1/$abottom), "the series \( \sum\limits_{n=1}^{\infty} $an \) converges by the Root Test.", 'convergent'],
   40   [ "\left(\frac{n}{n+$bbottom}\right)^n", "n", "n+$bbottom",  "=1", Real(1), "the Root Test is inconclusive for the series \( \sum\limits_{n=1}^{\infty} \left(\frac{n}{n+$bbottom}\right)^n \).  Because \[ \lim_{n\to\infty} a_n = \lim_{n\to\infty} \left( 1+\frac{$bbottom}{n}\right)^{-n} = \lim_{n\to\infty}\left[\left(1+\frac{$bbottom}{n} \right)^{n/$bbottom} \right]^{-$bbottom} =e^{-$bbottom} \ne 0,\] this series diverges by the Divergence Test.", 'The Root Test is inconclusive'] )};
   41 
   42 
   43 
   44 #Let's try to make a multiple choice question
   45 $question = new_multiple_choice();
   46 $question->qa(' \( \sum\limits_{n=1}^{\infty} $an \) is:', $trueanswer);
   47 $question->makeLast( 'convergent', 'divergent', 'The Root Test is inconclusive');
   48 
   49 
   50 
   51 
   52 Context()->texStrings;
   53 
   54 BEGIN_TEXT
   55 \{ beginproblem() \}
   56 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "39") \}
   57 $PAR
   58 Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive.
   59 \[  \sum\limits_{n=1}^{\infty} $an \]
   60 \(L = \lim\limits_{n \to \infty} \sqrt[n]{\left| a_n \right|} =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   61 $PAR
   62 \{ $question->print_q() \}
   63 \{ $question->print_a() \}
   64 END_TEXT
   65 
   66 
   67 Context()->normalStrings;
   68 
   69 #Answer Check Time!
   70 ANS($L->cmp);
   71 ANS(radio_cmp($question->correct_ans));
   72 
   73 Context()->texStrings;
   74 SOLUTION(EV3(<<'END_SOLUTION'));
   75 $PAR
   76 $SOL
   77 With \( a_n = $an \),
   78 \[ \sqrt[n]{a_n} = \sqrt[n]{$an} = \frac{$num}{$den} \] and \[ \lim_{n \to \infty} \sqrt[n]{a_n}  $result. \]
   79 
   80 Therefore, $FinalState
   81 
   82 END_SOLUTION
   83 
   84 ENDDOCUMENT()