[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.35.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

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Tue Nov 8 15:17:41 2011 UTC (2 years, 1 month ago) by aubreyja
File size: 1879 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', 'root test')
5 # TitleText1('Calculus: Early Transcendentals')
6 # EditionText1('2')
7 # AuthorText1('Rogawski')
8 # Section1('10.5')
9 # Problem1('35')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
16 #Load Necessary Macros
17
21
22
24
25 $ch = random(0,1,1); 26 27 if($ch>0){#convergent p
28    $p = random(2,9,1); 29$answer = "converges";
30 }else{
31    $p = random(.1,.9,.1); 32$answer = "diverges";
33 }
34
35
36 #Let's try to make a multiple choice question
37 $question = new_multiple_choice(); 38$question->qa("The $$p$$-series $$\sum\limits_{n=1}^{\infty} \frac{1}{n^{p}}$$ $answer by the Ratio Test.", 'False'); 39$question->makeLast('True', 'False');
40
41 Context()->texStrings;
42
43 BEGIN_TEXT
44 \{ beginproblem() \}
45 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "35") \}
46 $PAR 47 Determine if the following statment is True or False: 48$PAR
49 \{ $question->print_q() \} 50 \{$question->print_a() \}
51 END_TEXT
52
53 Context()->normalStrings;
54
55 #Answer Check Time!
56 ANS(radio_cmp($question->correct_ans)); 57 58 Context()->texStrings; 59 SOLUTION(EV3(<<'END_SOLUTION')); 60$PAR
61 \$SOL
62 With $$a_n = \frac{1}{n^{p}}$$,
63 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{(n+1)^{p}} \cdot \frac{n^{p}}{1} = \left( \frac{n}{n+1} \right)^{p} \quad \textrm{and} \quad \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1^{p} = 1.$
64 Therefore, the Ratio Test is inconclusive for the $$p$$-series  $$\sum\limits_{n=1}^\infty \frac1{n^{p}}$$.
65
66
67 END_SOLUTION
68
69 ENDDOCUMENT()


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