View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.29.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', 'ratio test')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.5')
    9 # Problem1('29')
   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 #$rho = 3
   27 
   28 $rho = random(2,9,1);
   29 $exp = random(2,9,1);
   30 
   31 $bn = "n^{$exp} a_n";
   32 
   33 #Let's try to make a multiple choice question
   34 $question = new_multiple_choice();
   35 $question->qa(' \( \sum\limits_{n=1}^{\infty} $bn \) is:', 'convergent');
   36 $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
   37 
   38 
   39 Context()->texStrings;
   40 
   41 BEGIN_TEXT
   42 \{ beginproblem() \}
   43 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "29") \}
   44 $PAR
   45 Assume that \( | \frac{a_{n+1}}{a_n}|\) converges to \( \rho = \frac{1}{$rho} \).  What can you say about the convergence of the given series?
   46 \[  \sum\limits_{n=1}^{\infty} {b_n} =  \sum\limits_{n=1}^{\infty} $bn\]
   47 \(\lim\limits_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).)
   48 $PAR
   49 \{ $question->print_q() \}
   50 \{ $question->print_a() \}
   51 END_TEXT
   52 
   53 Context()->normalStrings;
   54 
   55 #Answer Check Time!
   56 ANS(Real(1/$rho)->cmp);
   57 ANS(radio_cmp($question->correct_ans));
   58 
   59 Context()->texStrings;
   60 SOLUTION(EV3(<<'END_SOLUTION'));
   61 $PAR
   62 $SOL
   63 
   64 Let \( b_n = n^{$exp} a_n \).  Then
   65 
   66 \[ \rho =  \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right|
   67 = \lim_{n \to \infty} \left( \frac{n+1}{n} \right)^{$exp} \left| \frac{a_{n+1}}{a_n} \right| \] \[
   68  =  1^{$exp} \cdot \frac{1}{$rho} = \frac{1}{$rho} < 1\]
   69 Therefore, the series \( \sum\limits_{n=1}^{\infty} n^{$exp} a_n \) converges by the Ratio Test.
   70 
   71 
   72 END_SOLUTION
   73 
   74 ENDDOCUMENT()