View of /trunk/rochester_problib/instructiveProblems/NewProblems/benny_ratio2N.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('series', 'ratio test', 'absolute convergence')
    3 ##ENDDESCRIPTION
    4 
    5 DOCUMENT();        # This should be the first executable line in the problem.
    6 
    7 loadMacros(
    8 "PGstandard.pl",
    9 "PGcourse.pl",
   10 "MathObjects.pl",
   11 # "source.pl",
   12 );
   13 
   14 # No partial credit on this problem, so we say:
   15 install_problem_grader(~~&std_problem_grader);
   16 
   17 TEXT(beginproblem());
   18 $showPartialCorrectAnswers = 0;
   19 
   20 ###################
   21 #
   22 #  Setup
   23 
   24 Context()->variables->add(n => 'Real');
   25 Context()->strings->add(divergent=>{},div=>{alias=>'divergent'},
   26   diverges=>{alias=>'divergent'});
   27 
   28 $a = non_zero_random(-9,9,1);
   29 $b = random(1,9,1);
   30 $c = non_zero_random(-8,8,1);
   31 
   32 $expr_e  = Formula("e^(n + $a)");
   33 $expr_ne = Formula("e^(n + $a)*sqrt(n + $c)")->reduce;
   34 $expr_f  = Formula("(n + $b)!")->reduce;
   35 $expr_nf = Formula("sqrt(n + $c)*(n + $b)!")->reduce;
   36 
   37 $gen = random(0,3,1);
   38 
   39 Context()->texStrings;  #   Needed here to display the strings below correctly
   40                         #  in the main part of the problem.
   41 if ($gen == 0) {
   42   $frac = "\frac{$expr_e}{$expr_nf}";
   43   $soln1 = Real("0");
   44   $soln2 = Real("1");
   45   }
   46 
   47 elsif ($gen == 1) {
   48   $frac = "\frac{$expr_ne}{$expr_f}";
   49   $soln1 = Real("0");
   50   $soln2 = Real("1");
   51   }
   52 
   53 elsif ($gen == 2) {
   54   $frac = "\frac{$expr_nf}{$expr_e}";
   55   $soln1 = String("infinity");
   56   $soln2 = Real("2");
   57   }
   58 
   59 else {
   60   $frac = "\frac{$expr_f}{$expr_ne}";
   61   $soln1 = String("infinity");
   62   $soln2 = Real("2");
   63   }
   64 
   65 #### Prepare poplist for  question
   66 @num_of_sols_pop_up_list = (
   67     ' '       => '?',
   68        1   => "1. The Ratio Test says that the series converges absolutely.",
   69        2   => "2. The Ratio Test says that the series diverges.",
   70        3   => "3. The Ratio Test says that the series converges conditionally.",
   71        4   => "4. The Ratio Test is inconclusive, but the series converges
   72                absolutely by another test or tests.",
   73        5   => "5. The Ratio Test is inconclusive, but the series diverges
   74                by another test or tests.",
   75        6   => "6. The Ratio Test is inconclusive, but the series converges
   76                conditionally by another test or tests.",
   77 );
   78 
   79 ###################
   80 #
   81 #  Text
   82 
   83 BEGIN_TEXT
   84 
   85 Consider the series \($NewThing\)
   86 \( \displaystyle \sum_{n=1}^{\infty} a_n \)
   87 where
   88 \[ a_n = $frac \]
   89 In this problem you must attempt to use the Ratio Test to decide
   90 whether the series converges.
   91 
   92 $BR $BR
   93 
   94 Compute
   95 \[ L=\lim_{n\rightarrow\infty} \left| \frac{ a_{n+1} }{a_n} \right| \]
   96 Enter the numerical value of the limit L if it converges,
   97 INF if it diverges to infinity, -INF if it diverges to negative infinity,
   98 or DIV if it diverges but not to infinity or negative infinity. $BR
   99 \( L = \) \{ ans_rule( 30) \}
  100 
  101 $BR $BR
  102 
  103 Which of the following statements is true? $BR
  104 \{pop_up_list(@num_of_sols_pop_up_list)\}
  105 
  106 END_TEXT
  107 Context()->normalStrings;
  108 
  109 ###################
  110 #
  111 #  Custom Answer Checker
  112 
  113 $checker = sub {
  114 
  115    my ($correct, $student, $ans) = @_;
  116 
  117    Value::Error("Remember to take the absolute value of the limit.")
  118       if(($student < 0) ||($student == String("-infinity")));
  119 
  120    ($student == $correct);
  121 };
  122 
  123 ###################
  124 #
  125 #  Answers
  126 
  127 ANS($soln1->cmp(checker=>$checker));
  128 ANS($soln2->cmp);
  129 
  130 ENDDOCUMENT();        # This should be the last executable line in the problem.