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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2275 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    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('1')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #numerator = 1;
27 #denominator = 5^n
28
29 #Random Values
30 $base1 = Real(random(2, 9)); 31 32 33$denominator = "$base1^n"; 34$rho = 1/$base1; 35 36 ($series, $num,$den1, $den2,$trueanswer,) = @{list_random(
37   [ "\frac{1}{$denominator}", "1", "1", "$base1", 'convergent'],
38   [ "\frac{(-1)^{n-1}n}{$denominator} ", "n+1", "n", "$base1 n", 'convergent'])};
39
40
41
42
43 #Let's try to make a multiple choice question
44 $question = new_multiple_choice(); 45$question->qa(' $$\sum\limits_{n=1}^{\infty} series$$ is:', $trueanswer); 46$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
47
48
49 Context()->texStrings;
50
51 BEGIN_TEXT
52 \{ beginproblem() \}
53 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "1") \}
54 $PAR 55 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 56 $\sum\limits_{n=1}^{\infty} series$ 57 $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 58$PAR
59 \{ $question->print_q() \} 60 \{$question->print_a() \}
61 END_TEXT
62
63 Context()->normalStrings;
64
66 ANS($rho->cmp); 67 ANS(radio_cmp($question->correct_ans));
68
69 Context()->texStrings;
70 SOLUTION(EV3(<<'END_SOLUTION'));
71 $PAR 72$SOL
73 With $$a_n = series$$,
74 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num}{base1^{n+1}} \cdot \frac{base1^n}{den1} = \frac{num}{den2}$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{base1} < 1.$
75 Therefore, the series $$\sum\limits_{n=1}^{\infty} series$$ converges by the Ratio Test.
76
77
78 END_SOLUTION
79
80 ENDDOCUMENT()