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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/10.5.11.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2549 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('11')
10 # Author('LA Danielson')
11 # Institution('The College of Idaho')
12 DOCUMENT();
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25 $exp = Real(random(30,90,10)); 26 27 ($series, $num1,$den1, $num2,$den2, $result,$L, $rho,$compare, $answer,$trueanswer,) = @{list_random(
28   [ "\frac{e^n}{n!}", "e^{n+1}", "(n+1)!", "n!", "e^n", "\frac{e}{n+1}", 0, Real(0), "<", "converges", 'convergent'],
29   [ "\frac{e^n}{n^n} ", "e^{n+1}", "(n+1)^{n+1}", "n^n", ,"e^n","\frac{e}{n+1}\left( \frac{n}{n+1}\right)^n = \frac{e}{n+1}\left( 1+\frac{1}{n}\right)^{-n}", "0\cdot \frac{1}{e}=0", Real(0), "<", "converges", 'convergent'],
30         ["\frac{n^{$exp}}{n!}", "(n+1)^{$exp}", "(n+1)!", "n!", "n^{$exp}", "\frac{1}{n+1}\left( \frac{n+1}{n}\right)^{$exp}=\frac{1}{n+1}\left( 1+\frac{1}{n}\right)^{$exp}", "0\cdot 1=0", Real(0), "<", "converges", 'convergent' ])}; 31 32 33 34 35 36$question = new_multiple_choice();
37 $question->qa(' $$\sum_{n=1}^{\infty} series$$ is:', 'convergent'); 38$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
39
40
41 Context()->texStrings;
42
43 BEGIN_TEXT
44 \{ beginproblem() \}
45 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "11") \}
46 $PAR 47 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 48 $\sum\limits_{n=1}^{\infty} series$ 49 $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 50$PAR
51
52 \{ $question->print_q() \} 53 \{$question->print_a() \}
54 END_TEXT
55
56 Context()->normalStrings;
57
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59 ANS($rho->cmp); 60 ANS(radio_cmp($question->correct_ans));
61
62 Context()->texStrings;
63 SOLUTION(EV3(<<'END_SOLUTION'));
64 $PAR 65$SOL
66 With $$a_n = series$$,
67 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num1}{den1} \cdot \frac{num2}{den2} = result$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L compare 1.$
68 Therefore, the series $$\sum\limits_{n=1}^{\infty} series$$ \$answer by the Ratio Test.
69
70
71 END_SOLUTION
72
73 ENDDOCUMENT()