View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.19.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('19')
   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(3, 9, 1));
   24 
   25 $start = 1;
   26 
   27 $answer = "converges";
   28 
   29 $func = "\frac{1}{n$a^n}";
   30 
   31 $wrong = "converges";
   32 
   33 if ($answer eq "converges") {
   34     $wrong = "diverges";
   35 }
   36 
   37 $mc = new_multiple_choice();
   38 
   39 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   40     $answer);
   41 $mc->extra($wrong);
   42 $mc->makeLast("diverges");
   43 
   44 Context()->texStrings;
   45 BEGIN_TEXT
   46 \{ beginproblem() \}
   47 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","19") \}
   48 $PAR
   49 Use the Comparison Test to determine whether the infinite series is convergent.
   50 \[ \sum_{n=$start}^{\infty} $func \]
   51 
   52 By the Comparison Test,
   53 \{ $mc->print_q; \}
   54 \{ $mc->print_a; \}
   55 $BR
   56 $BBOLD Note: $EBOLD You are allowed only one attempt on this problem.
   57 $PAR
   58 END_TEXT
   59 Context()->normalStrings;
   60 
   61 ANS(str_cmp($mc->correct_ans));
   62 
   63 $j1 = 0;
   64 
   65 Context()->texStrings;
   66 SOLUTION(EV3(<<'END_SOLUTION'));
   67 $PAR
   68 $SOL
   69 $PAR
   70 We compare with the geometric series \( \displaystyle \sum \left(\frac{1}{$a}\right)^n \).  For \( n \ge 1 \),
   71 
   72 \[ $func \le \frac{1}{$a^n} = \left(\frac{1}{$a}\right)^n .\]
   73 
   74 Since \( \displaystyle \sum_{n=$start}^{\infty} \left(\frac{1}{$a}\right)^n \) converges (it's a geometric series with \( |r| = \frac{1}{$a} <1\) ), we conclude by the Comparison Test that \( \displaystyle \sum_{n=$start}^{\infty} $func \) also converges.
   75 $PAR
   76 END_SOLUTION
   77 
   78 ENDDOCUMENT();
   79 
   80