View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.56.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('56')
   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 #$constant = 12
   27 
   28 $constant = random(5, 35);
   29 
   30 
   31 #Let's try to make a multiple choice question
   32 $question = new_multiple_choice();
   33 $question->qa(' \( \sum\limits_{n=1}^{\infty} ( \frac{n}{n+$constant} )^n \) is:', 'divergent');
   34 $question->makeLast( 'convergent', 'divergent');
   35 
   36 
   37 Context()->texStrings;
   38 
   39 BEGIN_TEXT
   40 \{ beginproblem() \}
   41 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "56") \}
   42 $PAR
   43 Determine convergence or divergence using any method covered so far.
   44 $PAR
   45 \{ $question->print_q() \}
   46 \{ $question->print_a() \}
   47 END_TEXT
   48 
   49 Context()->normalStrings;
   50 
   51 #Answer Check Time!
   52 ANS(radio_cmp($question->correct_ans));
   53 
   54 Context()->texStrings;
   55 SOLUTION(EV3(<<'END_SOLUTION'));
   56 $PAR
   57 $SOL
   58 Because the general term has the form of a function of \(n\) raised to the \(n\)th power, we might be tempted to use the Root Test; however, the Root Test is inconclusive for this series.  Instead, note
   59 \[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 1 + \frac{$constant}{n} \right)^{-n} = \lim_{n \to \infty} \left[ \left( 1 + \frac{$constant}{n} \right)^{\frac{n}{$constant}} \right]^{-$constant} = e^{-$constant} \neq 0. \]
   60 Therefore, the series diverges by the Divergence Test.
   61 
   62 
   63 END_SOLUTION
   64 
   65 ENDDOCUMENT()