View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.25.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Infinite Series')
    3 ## DBsection('Convergence of Series with positive terms')
    4 ## KEYWORDS('calculus', 'infinite series', 'series', 'converge', 'convergence', 'comparison test', 'integral test', 'limit')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.3')
    9 ## Problem1('25')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("PGchoicemacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 $context = Context();
   19 
   20 $context->variables->add(n=>'Real');
   21 
   22 
   23 $a = Real(random(2, 6, 2));
   24 $b = list_random("\sin", "\cos");
   25 $np = Real(random(2, 9, 1));
   26 
   27 $start = 1;
   28 
   29 $answer = "converges";
   30 
   31 $func = "\frac{$b^{$a} n}{n^{$np}}";
   32 
   33 $wrong = "converges";
   34 
   35 if ($answer eq "converges") {
   36     $wrong = "diverges";
   37 }
   38 
   39 $mc = new_multiple_choice();
   40 
   41 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   42     $answer);
   43 $mc->extra($wrong);
   44 $mc->makeLast("diverges");
   45 
   46 Context()->texStrings;
   47 BEGIN_TEXT
   48 \{ beginproblem() \}
   49 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","25") \}
   50 $PAR
   51 Use the Comparison Test to determine whether the infinite series is convergent.
   52 \[ \sum_{n=$start}^{\infty} $func \]
   53 
   54 By the Comparison Test,
   55 
   56 \{ $mc->print_q; \}
   57 \{ $mc->print_a; \}
   58 $BR
   59 $BBOLD Note: $EBOLD You are allowed only one attempt on this problem.
   60 $PAR
   61 END_TEXT
   62 Context()->normalStrings;
   63 
   64 ANS(str_cmp($mc->correct_ans));
   65 
   66 $j1 = 0;
   67 
   68 Context()->texStrings;
   69 SOLUTION(EV3(<<'END_SOLUTION'));
   70 $PAR
   71 $SOL
   72 $PAR
   73 For \( n \ge 1 \), \( 0 \le $b^{$a} n \le 1 \), so
   74 
   75 \[ 0 \le $func \le \frac{1}{n^{$np}} .\]
   76 
   77 The series \( \displaystyle \sum_{n=$start}^{\infty} \frac{1}{n^{$np}} \) is a p-series with \( p = $np > 1 \), so it converges.  By the Comparison Test we can therefore conclude that series \(\sum\limits_{n=$start}^{\infty} $func\) also converges.
   78 $PAR
   79 END_SOLUTION
   80 
   81 ENDDOCUMENT();
   82 
   83