View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.48.pg

    1 # DBsubject('Calculus')
    2 # DBchapter('Infinite Series and Sequences')
    3 # DBsection('Absolute Convergence and the Root and Ratio Tests')
    4 # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'root test', 'ratio test')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.5')
    9 # Problem1('48')
   10 # Author('Emily Price')
   11 # Institution('W.H.Freeman')
   12 DOCUMENT();
   13 
   14 
   15 
   16 #Load Necessary Macros
   17 
   18 loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", );
   19 loadMacros("Parser.pl");
   20 loadMacros("freemanMacros.pl");
   21 
   22 
   23 Context()->variables->add(n=>'Real');
   24 
   25 #Book Values
   26 #numerator = n^2 + 4n
   27 #denominator = 3n^4 + 9
   28 
   29 #building the numerator - n^2 + bn + c
   30 $b1 = random(0, 9);
   31 $c1 = random(1, 9);
   32 $numerator = Formula("n^2 + $b1*n + $c1")->reduce;
   33 
   34 #building the denominator - an^4 + bn^3 + cn^2 + dn + e
   35 $a2 = random(2, 9);
   36 $b2 = random(0, 9);
   37 $c2 = random(0, 9);
   38 $d2 = random(1, 9);
   39 $e2 = random(0, 9);
   40 $denominator = Formula("$a2 n^4 + $b2 n^3 + $c2 n^2 + $d2 n + $c2")->reduce;
   41 
   42 #Let's try to make a multiple choice question
   43 $question = new_multiple_choice();
   44 $question->qa(' \( \sum\limits_{n=1}^{\infty} \frac{$numerator}{$denominator} \) is:', 'convergent');
   45 $question->makeLast( 'convergent', 'divergent');
   46 
   47 
   48 Context()->texStrings;
   49 
   50 BEGIN_TEXT
   51 \{ beginproblem() \}
   52 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "48") \}
   53 $PAR
   54 Determine convergence or divergence using any method covered so far.
   55 $PAR
   56 \{ $question->print_q() \}
   57 \{ $question->print_a() \}
   58 END_TEXT
   59 
   60 Context()->normalStrings;
   61 
   62 #Answer Check Time!
   63 ANS(radio_cmp($question->correct_ans));
   64 
   65 Context()->texStrings;
   66 SOLUTION(EV3(<<'END_SOLUTION'));
   67 $PAR
   68 $SOL
   69 
   70 This series is similar to a \( p \)-series; because
   71 \[ \frac{$numerator}{$denominator} \approx \frac{n^2}{$a2 n^4} = \frac{1}{$a2 n^2} \]
   72 for large \(n\), we will apply the Limit Comparison Test comparing with the \( p \)-series with \( p = 2 \).  Now,
   73 \[ L = \lim_{n \to \infty} \frac{\frac{$numerator}{$denominator}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{n^2($numerator)}{$denominator} = \frac{1}{$a2}. \]
   74 The \( p \)-series with \( p = 2 \) converges and \( L \) exists; therefore, the series also converges.
   75 
   76 END_SOLUTION
   77 
   78 ENDDOCUMENT()