View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.51.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('51')
   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 $a = Real(random(3, 7, 1));
   23 $b = Real(random(6, 9, 1));
   24 $p = Real(random(3, 7, 1));
   25 $pm1 = $p-1;
   26 $p2m1 = 2*$p-1;
   27 
   28 $start = 1;
   29 
   30 
   31 
   32 $ch = random(0,1,1); ##0 diverge, 1 converge! LAD
   33 
   34 if ($ch >0) {#converge
   35     $answer = "converges";
   36     $type = "convergent";
   37     $wrong = "diverges";
   38     $start = 1;
   39     $func = "\frac{\sqrt[$p]{n}}{$a n^2 + $b}";
   40     $bn = "\frac{1}{n^{\frac{$p2m1}{$p}}}";
   41     $cn = "\frac{n^2}{$a n^2 + $b}";
   42     $pnum = $p2m1;
   43     $compare = ">";
   44 
   45 }else{
   46     $answer = "diverges";
   47     $type = "divergent";
   48     $wrong = "converges";
   49     $start = 1;
   50     $func = "\frac{\sqrt[$p]{n}}{$a n + $b}";
   51     $bn = "\frac{1}{n^{\frac{$pm1}{$p}}}";
   52     $cn = "\frac{n}{$a n + $b}";
   53     $pnum = $pm1;
   54     $compare = "<";
   55 }
   56 
   57 
   58 $mc = new_multiple_choice();
   59 
   60 $mc->qa("Determine convergence or divergence of \( \displaystyle \sum_{n=$start}^{\infty} $func \).",
   61     $answer);
   62 $mc->extra($wrong);
   63 $mc->makeLast("diverges");
   64 
   65 Context()->texStrings;
   66 BEGIN_TEXT
   67 \{ beginproblem() \}
   68 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","51") \}
   69 $PAR
   70 \{ $mc->print_q; \}
   71 \{ $mc->print_a; \}
   72 $BR
   73 $BBOLD Note: $EBOLD You are allowed only one attempt on this problem.
   74 $PAR
   75 END_TEXT
   76 Context()->normalStrings;
   77 
   78 ANS(str_cmp($mc->correct_ans));
   79 
   80 $j1 = 0;
   81 
   82 Context()->texStrings;
   83 SOLUTION(EV3(<<'END_SOLUTION'));
   84 $PAR
   85 $SOL
   86 $PAR
   87 Apply the Limit Comparison Test with \( a_n = $func \) and \( b_n = $bn \):
   88 
   89 \[ L = \lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} \frac{$func}{$bn} = \lim_{n\to\infty} $cn = \frac{1}{$a} .\]
   90 
   91 The series \( \displaystyle \sum_{n=1}^{\infty} $bn \) is a $type \(p\)-series \(\left( p = \frac{$pnum}{$p} $compare 1\right)  \).  Because \( L > 0 \) exists, by the Limit Comparison Test we can conclude that the series \( \displaystyle \sum_{n=1}^{\infty} $func \) also $answer.
   92 $PAR
   93 END_SOLUTION
   94 
   95 ENDDOCUMENT();
   96 
   97