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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2472 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')
3 ## DBsection('Convergence of Series with positive terms')
4 ## KEYWORDS('calculus', 'infinite series', 'series', 'converge', 'convergence', 'comparison test', 'integral test', 'limit')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('10.3')
9 ## Problem1('21')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
18 $context = Context(); 19 20$context->variables->add(n=>'Real');
21
22
23 $a = Real(random(4, 9, 1)); 24$p = Real(random(2, 7, 1));
25
26 $ch = random(0,1,1); ##0 diverge, 1 converge! LAD 27 28 if ($ch >0) {#converge
29     $answer = "converges"; 30$wrong = "diverges";
31     $start = 1; 32$func = "\frac{n^{\frac{1}{$a}}}{n^{$p} + n}";
33     $bn = "\frac{n^{\frac{1}{$a}}}{n^{$p}}"; 34$b = $a *$p - 1;
35     $bounds = "\le"; 36$compare = ">";
37
38 }else{
39     $answer = "diverges"; 40$wrong = "converges";
41     $start = 2; 42$c = $a*($p-1)+1;
43     $func = "\frac{n^{\frac{$c}{$a}}}{n^{$p} - n}";
44     $bn = "\frac{n^{\frac{$c}{$a}}}{n^{$p}}";
45     $b =$a * $p -$c;
46     $bounds = "\ge"; 47$compare = "<";
48 }
49
50 $mc = new_multiple_choice(); 51 52$mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ",
53     $answer); 54$mc->extra($wrong); 55$mc->makeLast("diverges");
56
57 Context()->texStrings;
58 BEGIN_TEXT
59 \{ beginproblem() \}
60 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","21") \}
61 $PAR 62 Use the Comparison Test to determine whether the infinite series is convergent. 63 $\sum_{n=start}^{\infty} func$ 64 65 By the Comparison Test, 66 67 \{$mc->print_q; \}
68 \{ $mc->print_a; \} 69$BR
70 $BBOLD Note:$EBOLD You are allowed only one attempt on this problem.
71 $PAR 72 END_TEXT 73 Context()->normalStrings; 74 75 ANS(str_cmp($mc->correct_ans));
76
77 $j1 = 0; 78 79 Context()->texStrings; 80 SOLUTION(EV3(<<'END_SOLUTION')); 81$PAR
82 $SOL 83$PAR
84 For $$n \ge start$$,
85
86 $func bounds bn = \frac{1}{n^{\frac{b}{a}}} .$
87
88 The series $$\displaystyle \sum_{n=start}^{\infty} \frac{1}{n^{\frac{b}{a}}}$$ is a $$p$$-series with $$p = \frac{b}{a} compare 1$$, so it $answer. By the Comparison Test we can therefore conclude that the series $$\sum_{n=start}^{\infty} func$$ also$answer.
89 \$PAR
90 END_SOLUTION
91
92 ENDDOCUMENT();
93
94