[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.31.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/10.5.31.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 1 month ago) by aubreyja
File size: 2552 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('31')
10 # Author('LA Danielson')
11 # Institution('The College of Idaho')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #$rho = 1/3, 3^n a_n 27 28$den = random(2,9,1);
29 $base = random(2,9,1); 30$rho = Real($base/$den);
31 $display = "\frac{$base}{$den}"; 32 33$bn = "$base^{n} a_n"; 34 35 if($den==$base){ 36$trueanswer = 'The Ratio Test is inconclusive';
37    $compare = "="; 38$FinalState = "the Ratio Test is inconclusive for the series $$\sum\limits_{n=1}^{\infty} bn .$$";
39 }elsif($den>$base){
40    $trueanswer = 'convergent'; 41$compare = "<";
42    $FinalState = "the series $$\sum\limits_{n=1}^{\infty} bn$$ converges by the Ratio Test."; 43 }else{ 44$trueanswer = 'divergent';
45    $compare = ">"; 46$FinalState = "the series $$\sum\limits_{n=1}^{\infty} bn$$ diverges by the Ratio Test.";
47 }
48
49 $question = new_multiple_choice(); 50$question->qa(' $$\sum\limits_{n=1}^{\infty} bn$$ is:', $trueanswer); 51$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
52
53
54 Context()->texStrings;
55
56 BEGIN_TEXT
57 \{ beginproblem() \}
58 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "31") \}
59 $PAR 60 Assume that $$| \frac{a_{n+1}}{a_n}|$$ converges to $$\rho = \frac{1}{den}$$. What can you say about the convergence of the given series? 61 $\sum\limits_{n=1}^{\infty} {b_n} = \sum\limits_{n=1}^{\infty} bn$ 62 $$\lim\limits_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 63$PAR
64 \{ $question->print_q() \} 65 \{$question->print_a() \}
66 END_TEXT
67
68 Context()->normalStrings;
69
71 ANS($rho->cmp); 72 ANS(radio_cmp($question->correct_ans));
73
74 Context()->texStrings;
75 SOLUTION(EV3(<<'END_SOLUTION'));
76 $PAR 77$SOL
78 Let $$b_n = bn$$.  Then
79 $\rho = \lim_{n \to \infty} \left| \frac{b_{n+1}}{b_n} \right| = \lim_{n \to \infty} \frac{base^{n+1}}{base^n} \left| \frac{a_{n+1}}{a_n} \right| 80 = base \cdot \frac{1}{den} = display compare 1.$
81 Therefore, \$FinalState
82
83
84 END_SOLUTION
85
86 ENDDOCUMENT()