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# View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.17.pg

Tue Nov 8 15:17:41 2011 UTC (19 months, 1 week ago) by aubreyja
File size: 2429 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('17')
10 # Author('LA Danielson')
11 # Institution('The College of Idaho')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #numerator = n^2
27 #denominator = (2n + 1)!
28
29 $a = random(2,9,1); 30 31 32 ($series, $num1,$den1, $num2,$den2, $result,$L, $rho,$compare, $answer,$trueanswer,) = @{list_random(
33   [ "\frac{(n!)^2}{(2n)!}", "((n+1)!)^2", "(2n+2)!", "(2n)!", "(n!)^2", "\frac{(n+1)^2}{(2n+2)(2n+1)}", "\frac{1}{4}", Real(1/4), "<", "converges", 'convergent'],
34   [ "\frac{$a n^2}{(2n+1)!}", "$a (n+1)^2",  "(2n+3)!",  "(2n+1)!", "$a n^2", "\left(\frac{n+1}{n} \right)^2 \frac{1}{(2n+3)(2n+2)}", "1^2\cdot 0 = 0", Real(0), "<", "converges", 'convergent'] )}; 35 36 37 38 39 #Let's try to make a multiple choice question 40$question = new_multiple_choice();
41 $question->qa(' $$\sum_{n=1}^{\infty} series$$ is:', 'convergent'); 42$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
43
44
45 Context()->texStrings;
46
47 BEGIN_TEXT
48 \{ beginproblem() \}
49 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "17") \}
50 $PAR 51 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 52 $\sum\limits_{n=1}^{\infty} series$ 53 $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 54$PAR
55
56 \{ $question->print_q() \} 57 \{$question->print_a() \}
58 END_TEXT
59
60 Context()->normalStrings;
61
62 ANS($rho->cmp); 63 ANS(radio_cmp($question->correct_ans));
64
65 Context()->texStrings;
66 SOLUTION(EV3(<<'END_SOLUTION'));
67 $PAR 68$SOL
69
70 With $$a_n = series$$,
71 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num1}{den1} \cdot \frac{num2}{den2} = result$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L compare 1.$
72 Therefore, the series $$\sum\limits_{n=1}^{\infty} series$$ \$answer by the Ratio Test.
73
74
75
76 END_SOLUTION
77
78 ENDDOCUMENT()