View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.37.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('37')
   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 #$rho = 1
   27 
   28 $exp = random(2, 9);
   29 $base = random(5,15,1);
   30 
   31 ($an, $den, $result, $L, $compare, $answer, $trueanswer,) = @{list_random(
   32   [ "\frac{1}{n^{$exp n}}", "n^{$exp}", "0", Real(0), "<", "converges", 'convergent'],
   33   [ "\frac{1}{$base^n}", "$base",  "\frac{1}{$base}", Real(1/$base), "<", "converges", 'convergent'] )};
   34 
   35 
   36 
   37 #Let's try to make a multiple choice question
   38 $question = new_multiple_choice();
   39 $question->qa(' \( \sum\limits_{n=1}^{\infty} $an \) is:', $trueanswer);
   40 $question->makeLast( 'convergent', 'divergent', 'The Root Test is inconclusive');
   41 
   42 
   43 Context()->texStrings;
   44 
   45 BEGIN_TEXT
   46 \{ beginproblem() \}
   47 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "37") \}
   48 $PAR
   49 Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive.
   50 \[  \sum\limits_{n=1}^{\infty} $an \]
   51 \(L = \lim\limits_{n \to \infty} \sqrt[n]{\left| a_n \right|} =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   52 $PAR
   53 \{ $question->print_q() \}
   54 \{ $question->print_a() \}
   55 END_TEXT
   56 
   57 Context()->normalStrings;
   58 
   59 #Answer Check Time!
   60 ANS($L->cmp);
   61 ANS(radio_cmp($question->correct_ans));
   62 
   63 Context()->texStrings;
   64 SOLUTION(EV3(<<'END_SOLUTION'));
   65 $PAR
   66 $SOL
   67 With \( a_n = $an \),
   68 \[ \sqrt[n]{a_n} = \sqrt[n]{$an} = \frac{1}{$den} \] and \[ L = \lim_{n \to \infty} \sqrt[n]{a_n} = $result $compare 1. \]
   69 Therefore, the series \( \sum\limits_{n=1}^{\infty} $an \) $answer by the Root Test.
   70 
   71 END_SOLUTION
   72 
   73 ENDDOCUMENT()