View of /trunk/NationalProblemLibrary/Rochester/setSeries7AbsolutelyConvergent/benny_root1.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('series','divergent','convergent','absolute convergence','root test','conditionally convergent')
    6 ## Tagged by cmd6a 5/6/06
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Infinite Sequences and Series')
   10 ## DBsection('Absolute Convergence and the Ratio and Root Tests')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 ## TitleText2('Calculus: Early Transcendentals')
   20 ## EditionText2('1')
   21 ## AuthorText2('Rogawski')
   22 ## Section2('10.5')
   23 ## Problem2('37')
   24 
   25 DOCUMENT();        # This should be the first executable line in the problem.
   26 
   27 loadMacros(
   28 "PG.pl",
   29 "PGbasicmacros.pl",
   30 "PGchoicemacros.pl",
   31 "PGanswermacros.pl",
   32 "PGauxiliaryFunctions.pl"
   33 );
   34 
   35 # No partial credit on this problem, so we say:
   36 install_problem_grader(~~&std_problem_grader);
   37 
   38 
   39 TEXT(beginproblem());
   40 $showPartialCorrectAnswers = 0;
   41 
   42 $a_numb = random(1,5,1);
   43 $a_sign = random(-1,1,2);
   44 $a      = $a_numb * $a_sign;
   45 if ($a == 1) {$coef_a = ""};
   46 if ($a == -1) {$coef_a = "-"};
   47 if ($a_numb > 1) {$coef_a = $a};
   48 
   49 $b_numb = random(1,9,1);
   50 $b_sign = random(-1,1,2);
   51 $b      = $b_numb * $b_sign;
   52 
   53 $c_numb = $a_numb;
   54 while ($c_numb == $a_numb) {$c_numb = random(1,5,1)};
   55 $c_sign = random(-1,1,2);
   56 $c      = $c_numb * $c_sign;
   57 if ($c == 1) {$coef_c = ""};
   58 if ($c == -1) {$coef_c = "-"};
   59 if ($c_numb > 1) {$coef_c = $c};
   60 
   61 $d_numb = random(1,9,1);
   62 $temp   = $a_numb * $b_numb;
   63 if ($temp == 1) { $d_sign = -$c_sign }
   64 else { $d_sign = $c_sign };
   65 $d      = $d_numb * $d_sign;
   66 
   67 $exp_num = random(1,2,1);
   68 if ($exp_num == 1) {$coef_en = "n"}
   69 else {$coef_en = "$exp_num n"};
   70 
   71 $exp_den = random(1,2,1);
   72 if ($exp_den == 1) {$coef_ed = "n"}
   73 else {$coef_ed = "$exp_den n"};
   74 
   75 if ($exp_num > $exp_den) {
   76   $soln1 = 'INF';
   77   $soln2 = 'B';
   78   };
   79 if ($exp_num < $exp_den) {
   80   $soln1 = 0;
   81   $soln2 = 'A';
   82   };
   83 if ($exp_num == $exp_den) {
   84   $soln1 = ($a_numb/$c_numb)**$exp_num;
   85   if ($soln1 < 1) {$soln2 = 'A'}
   86   else {$soln2 = 'B'};
   87   };
   88 
   89 $numerator = "($coef_a n + $b)^{$coef_en}";
   90 $denominator = "($coef_c n + $d)^{$coef_ed}";
   91 
   92 $frac = "\frac{ $numerator }{ $denominator }";
   93 
   94 TEXT(EV2(<<EOT));
   95 
   96 Consider the series
   97 \( \displaystyle \sum_{n=1}^{\infty} a_n \)
   98 where
   99 \[ a_n = $frac \]
  100 In this problem you must attempt to use the Root Test to decide
  101 whether the series converges.
  102 
  103 $BR $BR
  104 
  105 Compute
  106 \[ L=\lim_{n\rightarrow\infty} \sqrt[n]{|a_n|} \]
  107 Enter the numerical value of the limit L if it converges,
  108 INF if it diverges to infinity, MINF if it diverges to negative infinity,
  109 or DIV if it diverges but not to infinity or negative infinity. $BR
  110 \( L = \) \{ans_rule( 30) \}
  111 
  112 $BR $BR
  113 
  114 Which of the following statements is true? $BR
  115 A. The Root Test says that the series converges absolutely. $BR
  116 B. The Root Test says that the series diverges. $BR
  117 C. The Root Test says that the series converges conditionally. $BR
  118 D. The Root Test is inconclusive, but the series converges
  119   absolutely by another test or tests. $BR
  120 E. The Root Test is inconclusive, but the series diverges
  121   by another test or tests. $BR
  122 F. The Root Test is inconclusive, but the series converges
  123   conditionally by another test or tests. $BR
  124 Enter the letter for your choice here:
  125 \{ans_rule( 10) \}
  126 
  127 EOT
  128 ANS(num_cmp($soln1, strings=>['INF', 'MINF', 'DIV']));
  129 ANS(str_cmp($soln2));
  130 
  131 ENDDOCUMENT();        # This should be the last executable line in the problem.