[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.5.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/10.5.5.pg

Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2404 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('5')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 # $exp1 = 2 27 #$exp2 = 1
28 # $constant = 1 29 30$exp1 = random(2, 7);
31 $exp2 = random(1,$exp1-1,1);#Formula("$exp1 - 1")->reduce; 32$constant = random(1, 9);
33 $num = Formula("n^{$exp2}")->reduce;
34 $num2 = Formula("(n+1)^{$exp2}")->reduce;
35
36 $denominator = "n^{$exp1} + $constant"; 37$denomplus1 = "(n+1)^{$exp1} +$constant";
38
39 $rho = Real(1); 40 41 #Let's try to make a multiple choice question 42$question = new_multiple_choice();
43 $question->qa(' $$\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$$ is:', 'The Ratio Test is inconclusive'); 44$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
45
46
47 Context()->texStrings;
48
49 BEGIN_TEXT
50 \{ beginproblem() \}
51 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "5") \}
52 $PAR 53 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 54 $\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$ 55 $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 56$PAR
57 \{ $question->print_q() \} 58 \{$question->print_a() \}
59 END_TEXT
60
61 Context()->normalStrings;
62
64 ANS($rho->cmp); 65 ANS(radio_cmp($question->correct_ans));
66
67 Context()->texStrings;
68 SOLUTION(EV3(<<'END_SOLUTION'));
69 $PAR 70$SOL
71 With $$a_n = \frac{num}{denominator}$$,
72
73 $\left| \frac{a_{n+1}}{a_n} \right| = 74 \frac{num2}{denomplus1} \cdot \frac{denominator}{num} = \frac{num2}{num} \cdot \frac{denominator}{denomplus1},$
75 and
76 $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \cdot 1 = 1.$
77 Therefore, for the series $$\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$$, the Ratio Test is inconclusive.
78
79 END_SOLUTION
80
81 ENDDOCUMENT()