[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.29.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# 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

Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2100 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('29')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
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22
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
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()