View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.3.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('3')
   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, 9, 1));
   24 $a2 = $a - 1;
   25 $a3 = Real(1/$a2);
   26 $b = Real(random(2,7,1));
   27 $bm1 = $b-1;
   28 $bp1 = $b+1;
   29 
   30 ($func, $f, $ans_lim, $func2, $lim, $answer) = @{ list_random(
   31 ["n^{-\frac{1}{$b}}", Formula(" x^(-1/$b)") , "inf"  , "x^{-\frac{1}{$b}}", "\(\frac{$b}{$bm1} \displaystyle \lim_{R\to\infty}\left(R^{\frac{$bm1}{$b}} - 1\right) = \infty \)", "diverges"],
   32 ["n^{-\frac{$b}{$bp1}}", Formula("x^(-$b/$bp1) "), "inf","x^{-\frac{$b}{$bp1}}", "\($bp1 \displaystyle \lim_{R\to\infty}\left(R^{\frac{1}{$bp1}} - 1\right) = \infty \)", "diverges"],
   33 ["n^{-$a}", Formula(" x^(-$a)"), $a3 ,"x^{-$a}", "\(-\frac{1}{$a2} \displaystyle \lim_{R\to\infty}\left(R^{-$a2} - 1\right) = \frac{1}{$a2} \)", "converges"]
   34 ) };
   35 
   36 $start = 1;
   37 
   38 $wrong = "converges";
   39 
   40 if ($answer eq "converges") {
   41     $wrong = "diverges";
   42 }
   43 
   44 $mc = new_multiple_choice();
   45 
   46 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   47     $answer);
   48 $mc->extra($wrong);
   49 $mc->makeLast("diverges");
   50 
   51 Context()->texStrings;
   52 BEGIN_TEXT
   53 \{ beginproblem() \}
   54 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","3") \}
   55 $PAR
   56 Use the Integral Test to determine whether the infinite series is convergent.
   57 \[ \sum_{n=$start}^{\infty} $func \]
   58 Fill in the corresponding integrand and
   59 the value of the improper integral.
   60 $BR
   61 Enter $BBOLD inf $EBOLD for \(\infty\), $BBOLD -inf $EBOLD for \(-\infty\),
   62 and $BBOLD DNE $EBOLD if the limit does not exist.
   63 $PAR
   64 Compare with
   65 \(\int_{$start}^{\infty} \) \{ ans_rule() \} \(dx\) = \{ ans_rule() \}
   66 $PAR
   67 By the Integral Test,
   68 $BR
   69 
   70 \{ $mc->print_q; \}
   71 \{ $mc->print_a; \}
   72 $PAR
   73 END_TEXT
   74 Context()->normalStrings;
   75 
   76 
   77 ANS($f->cmp);
   78 ANS( num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"]));
   79 ANS(str_cmp($mc->correct_ans));
   80 
   81 $j1 = 0;
   82 
   83 Context()->texStrings;
   84 SOLUTION(EV3(<<'END_SOLUTION'));
   85 $PAR
   86 $SOL
   87 $PAR
   88 Let \( f(x) = $func2 \).  This function is continuous, positive and decreasing on the interval \( x \ge $start \), so the Integral Test applies.  Moreover,
   89 $PAR
   90 \( \displaystyle \int_{$start}^{\infty} $func2 \, dx = \displaystyle \lim_{R\to\infty} \displaystyle \int_{$start}^{R} $func2 \, dx = \) $lim.
   91 $PAR
   92 The integral $answer; hence the series \( \displaystyle \sum_{n=$start}^{\infty} $func \) also $answer.
   93 $PAR
   94 END_SOLUTION
   95 
   96 ENDDOCUMENT();
   97 
   98