[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.53.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/10.5.53.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 4 months ago) by aubreyja
File size: 2235 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('')
3 # DBsection('')
4 # KEYWORDS('')
5 # TitleText1('Calculus: Early Transcendentals')
6 # EditionText1('2')
7 # AuthorText1('Rogawski')
8 # Section1('10.5')
9 # Problem1('53')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #sin (1/n^2)
27
28 $exp = random(2, 9); 29 30$denominator = "n^{$exp}"; 31 32$sol1 = "Because $\lim_{n\to\infty} \cos\left(\frac{1}{denominator}\right) = \cos 0 = 1 \ne 0,$ the general term in the series $$\sum\limits_{n=1}^{\infty} (-1)^n \cos\left(\frac{1}{denominator}\right)$$ does not tend toward zero; therefore, the series diverges by the Divergence Test.";
33
34 $sol2 = "Here, we will apply the Limit Comparison Test, comparing with the $$p$$-series with $$p = exp$$. Now, 35 $L = \lim_{n \to \infty} \frac{\sin \left(\frac{1}{denominator}\right)}{\frac{1}{n^{exp}}} = \lim_{u \to 0} \frac{\sin u}{u} = 1,$ 36 where $$u = \frac{1}{n^{exp}}$$. The $$p$$-series with $$p = exp$$ converges and $$L$$ exists; therefore, the series $$\sum\limits_{n=1}^{\infty} \sin \left(\frac{1}{denominator}\right)$$ also converges."; 37 38 ($an, $solution,$trueanswer) = @{list_random(
39   [ "(-1)^n \cos \left(\frac{1}{$denominator}\right)",$sol1, 'divergent'],
40   [ "\sin \left(\frac{1}{$denominator}\right)",$sol2, 'convergent'] )};
41
42
43 #Let's try to make a multiple choice question
44 $question = new_multiple_choice(); 45$question->qa(' $$\sum\limits_{n=1}^{\infty} an$$ is:', $trueanswer); 46$question->makeLast( 'convergent', 'divergent');
47
48
49 Context()->texStrings;
50
51 BEGIN_TEXT
52 \{ beginproblem() \}
53 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "53") \}
54 $PAR 55 Determine convergence or divergence using any method covered so far. 56$PAR
57 \{ $question->print_q() \} 58 \{$question->print_a() \}
59 END_TEXT
60
61 Context()->normalStrings;
62
64 ANS(radio_cmp($question->correct_ans)); 65 66 Context()->texStrings; 67 SOLUTION(EV3(<<'END_SOLUTION')); 68$PAR
69 $SOL 70$solution