View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.3_Convergence_of_Series_with_Positive_Terms/10.3.5.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('5')
   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 $start = Real(random(10, 25, 1));
   23 $pc = Real(random(3, 9, 1));
   24 $a =  Real(random(3, 9, 2));
   25 $am2 = $a-2;
   26 $ap2 = $a+2;
   27 $pc4 = 4*$pc;
   28 $a3m4 = 3*$a-4;
   29 $a3m2 = 3*$am2;
   30 $spc = $start**3+$pc;
   31 
   32 
   33 $int_lim = Real(2/(3*$am2*($spc)**($am2/2)));
   34 
   35 ($func, $f, $ans_lim, $func2, $deriv, $lim, $answer) = @{ list_random(
   36 ["\frac{n^2}{\left(n^3 + $pc\right)^{\frac{$a}{2}}}",
   37 Formula("x^2/(x^3 + $pc)^($a/2)"),
   38 $int_lim ,
   39 "\frac{x^2}{\left(x^3 + $pc\right)^{\frac{$a}{2}}}",
   40 "\frac{2x\left(x^3+$pc \right)^{\frac{$a}{2}}-x^2\cdot \frac{$a}{2}\left( x^3+$pc \right)^{\frac{$am2}{2}}\cdot 3x^2}{\left(x^3+$pc \right)^{$a}} = \frac{x($pc4-$a3m4 x^3)}{2(x^3+$pc)^{\frac{$ap2}{2}}}",
   41 "-\frac{2}{$a3m2} \lim_{R\to\infty} \left(\frac{1}{(R^3 + $pc)^{\frac{$am2}{2}}} - \frac{1}{$spc^{\frac{$am2}{2}}}\right) = \frac{2}{$a3m2 \cdot $spc^{\frac{$am2}{2}}} ",
   42 "converges"]
   43 ) };
   44 
   45 $extra = "To evaluate the improper integral, we use the substitution \( u = x^3 + $pc \), \( du = 3x^2 \, dx \).  We then find$PAR";
   46 
   47 $wrong = "converges";
   48 
   49 if ($answer eq "converges") {
   50     $wrong = "diverges";
   51 }
   52 
   53 $mc = new_multiple_choice();
   54 
   55 $mc->qa("the infinite series \( \displaystyle \sum_{n=$start}^{\infty} $func \) ",
   56     $answer);
   57 $mc->extra($wrong);
   58 $mc->makeLast("diverges");
   59 
   60 Context()->texStrings;
   61 BEGIN_TEXT
   62 \{ beginproblem() \}
   63 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","5") \}
   64 $PAR
   65 Use the Integral Test to determine whether the infinite series is convergent.
   66 \[ \sum_{n=$start}^{\infty} $func \]
   67 Fill in the corresponding integrand and
   68 the value of the improper integral.
   69 $BR
   70 Enter $BBOLD inf $EBOLD for \(\infty\), $BBOLD -inf $EBOLD for \(-\infty\),
   71 and $BBOLD DNE $EBOLD if the limit does not exist.
   72 $PAR
   73 Compare with
   74 \(\int_{$start}^{\infty} \) \{ ans_rule() \} \(dx\) = \{ ans_rule() \}
   75 $PAR
   76 By the Integral Test,
   77 $BR
   78 \{ $mc->print_q; \}
   79 \{ $mc->print_a; \}
   80 $PAR
   81 END_TEXT
   82 Context()->normalStrings;
   83 
   84 ANS($f->cmp);
   85 ANS(num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"]));
   86 ANS(str_cmp($mc->correct_ans));
   87 
   88 Context()->texStrings;
   89 SOLUTION(EV3(<<'END_SOLUTION'));
   90 $PAR
   91 $SOL
   92 $PAR
   93 Let \( f(x) = $func2 \).  This function is continuous and positive on the interval \( x \ge $start \).  Moreover, because
   94 
   95 \[ f'(x) = $deriv, \]
   96 
   97 we see that \( f'(x) < 0 \) for \( x \ge $start \), so f is decreasing on the interval \( x \ge $start \).  The Integral Test therefore applies.  $extra
   98 $PAR
   99 \[ \int_{$start}^{\infty} $func2 \, dx =  \lim_{R\to\infty} \int_{$start}^{R} $func2 \, dx = \frac{1}{3}\lim_{R\to\infty}\int_{$start^3 + $pc}^{R^3 +$pc} \frac{du}{u^{$a/2}} \]
  100 \[= $lim. \]
  101 $PAR
  102 The integral $answer; hence the series \( \displaystyle \sum_{n=$start}^{\infty} $func \) also $answer.
  103 $PAR
  104 END_SOLUTION
  105 
  106 ENDDOCUMENT();
  107 
  108