[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.7.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/10.5.7.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2589 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('7')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 # $exp = 100 27 #$base = 2
28
29 $exp = random(100, 200); 30$base = random(2, 20);
31 $a = random(3, 9,1); #LAD added for 10.5.8 32$am1 = $a-1; 33 34 ($series, $num1,$den1, $num2,$den2, $result,$L, $rho,$compare, $answer,$trueanswer,) = @{list_random(
35   [ "\frac{$base^n}{n^{$exp}}","$base^{n+1}", "(n+1)^{$exp}", "n^{$exp}", "$base^n", "$base \left(\frac{n}{n+1}\right)^{$exp}", "$base\cdot 1^{$exp} = $base", Real($base), ">", "diverges", 'divergent'],
36   [ "\frac{n^{$a}}{$a^{n^{$am1}}} ", "(n+1)^{$a}", "$a^{(n+1)^{$am1}}", "$a^{n^{$am1}}", "n^{$a}","\left( \frac{n+1}{n}\right)^{$a}\cdot\frac{1}{$a^{(n+1)^{$am1}-n^{$am1}}} ","1^{$a}\cdot 0=0", Real(0), "<", "converges", 'convergent'])};
37
38
39
40 #Let's try to make a multiple choice question
41 $question = new_multiple_choice(); 42$question->qa(' $$\sum\limits_{n=1}^{\infty}series$$ is:', $trueanswer); 43$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive');
44
45
46 Context()->texStrings;
47
48 BEGIN_TEXT
49 \{ beginproblem() \}
50 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "7") \}
51 $PAR 52 Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 53 $\sum\limits_{n=1}^{\infty} series$ 54 $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.) 55$PAR
56 \{ $question->print_q() \} 57 \{$question->print_a() \}
58 END_TEXT
59
60 Context()->normalStrings;
61
63 ANS($rho->cmp); 64 ANS(radio_cmp($question->correct_ans));
65
66 Context()->texStrings;
67 SOLUTION(EV3(<<'END_SOLUTION'));
68 $PAR 69$SOL
70 With $$a_n = series$$,
71 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num1}{den1} \cdot \frac{num2}{den2} = result$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L compare 1.$
72 Therefore, the series $$\sum\limits_{n=1}^{\infty} series$$ \$answer by the Ratio Test.
73
74
75
76
77 END_SOLUTION
78
79 ENDDOCUMENT()