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    1 # DBsubject('Calculus')
    2 # DBchapter('Infinite Series and Sequences')
    3 # DBsection('Absolute Convergence and the Root and Ratio Tests')
    4 # KEYWORDS('calculus', 'series', 'sequences', 'alternating series', 'convergence', 'absolute convergence')
    5 # TitleText1('Calculus: Early Transcendentals')
    6 # EditionText1('2')
    7 # AuthorText1('Rogawski')
    8 # Section1('10.4')
    9 # Problem1('19')
   10 # Author('LA Danielson')
   11 # Institution('The College of Idaho')
   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 #Book Values
   23 #$series = \sum 1/3^n+5^n #19c, also add in #20d,21c,22d,23c
   24 
   25 $a = random(3,7,1);
   26 $b = $a+2;
   27 $am1 = $a-1;
   28 
   29 
   30 $an1 = "\frac{1}{$a^n+$b^n}";
   31 $an2 = "\frac{n^{$am1}}{n^{$a}-n}";
   32 $an3 = "\frac{(-1)^n}{\sqrt{n^2+$a}}";
   33 $an4 = "\frac{1}{\sqrt{n^2+$a}}";
   34 
   35 $hn = "\frac{1}{n}";
   36 
   37 #19c
   38 $sol1 = "For \(n\ge 1\),
   39 
   40 \[  $an1 \le \frac{1}{$a^n}=\left(\frac{1}{$a} \right)^n. \]
   41 
   42 The series \(\sum\limits_{n=1}^{\infty} \left(\frac{1}{$a} \right)^n\) is a convergent geometric series, so the Comparison Test implies that the series \(\sum\limits_{n=1}^{\infty} $an1\) converges.";
   43 
   44 #20d
   45 $sol2 = "Apply the Limit Comparison Test and compare with the divergent harmonic series:
   46 
   47 \[ L=\lim_{n\to\infty}\frac{$an2}{$hn} =
   48 \lim_{n\to\infty} \frac{n^{$a}}{n^{$a}-n}=1 .\]
   49 Because \(L>0\), we conclude that the series \( \sum\limits_{n=1}^{\infty} $an2 \) diverges.";
   50 
   51 #21c
   52 $sol3 = "This is an alternating series with \(a_n = $an3\).  Because \(a_n\) is a decreasing sequence that converges to zero, the series \( \sum\limits_{n=1}^{\infty} $an3 \) converges by the Leibniz Test.";
   53 
   54 #22d
   55 $sol4 = "Apply the Limit Comparison Test and compare with the divergent harmonic series:
   56 
   57 \[ L=\lim_{n\to\infty}\frac{$an4}{$hn} =
   58 \lim_{n\to\infty} \frac{n}{\sqrt{n^2+$a}}=1 .\]
   59 Because \(L>0\), we conclude that the series \( \sum\limits_{n=1}^{\infty} $an4 \) diverges.";
   60 
   61 ($series, $trueanswer, $solution) = @{list_random(
   62   [ "\( \sum\limits_{n=1}^{\infty} $an1 \)", 'converges',$sol1],
   63   [ "\( \sum\limits_{n=1}^{\infty} $an2 \)", 'diverges',$sol2],
   64         [ "\( \sum\limits_{n=1}^{\infty} $an3 \)", 'converges',$sol3],
   65         [ "\( \sum\limits_{n=1}^{\infty} $an4 \)", 'diverges',$sol4],
   66 
   67 )};
   68 
   69 
   70 
   71 #make a multiple choice question
   72 $question = new_multiple_choice();
   73 $question->qa(' $series ', $trueanswer);
   74 $question->makeLast( 'converges', 'diverges');
   75 
   76 
   77 Context()->texStrings;
   78 
   79 BEGIN_TEXT
   80 \{ beginproblem() \}
   81 \{ textbook_ref_exact("Rogawski ET 2e", "10.4", "19") \}
   82 $PAR
   83 Determine convergence or divergence by any method.
   84 $PAR
   85 \{ $question->print_q() \}
   86 \{ $question->print_a() \}
   87 END_TEXT
   88 
   89 Context()->normalStrings;
   90 
   91 #Answer Check Time!
   92 ANS(radio_cmp($question->correct_ans));
   93 
   94 Context()->texStrings;
   95 SOLUTION(EV3(<<'END_SOLUTION'));
   96 $PAR
   97 $SOL
   98 
   99 $solution
  100 
  101 END_SOLUTION
  102 
  103 ENDDOCUMENT()