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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2073 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('25')
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 $a = Real(random(2, 6, 2)); 24$b = list_random("\sin", "\cos");
25 $np = Real(random(2, 9, 1)); 26 27$start = 1;
28
29 $answer = "converges"; 30 31$func = "\frac{$b^{$a} n}{n^{$np}}"; 32 33$wrong = "converges";
34
35 if ($answer eq "converges") { 36$wrong = "diverges";
37 }
38
39 $mc = new_multiple_choice(); 40 41$mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ",
42     $answer); 43$mc->extra($wrong); 44$mc->makeLast("diverges");
45
46 Context()->texStrings;
47 BEGIN_TEXT
48 \{ beginproblem() \}
49 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","25") \}
50 $PAR 51 Use the Comparison Test to determine whether the infinite series is convergent. 52 $\sum_{n=start}^{\infty} func$ 53 54 By the Comparison Test, 55 56 \{$mc->print_q; \}
57 \{ $mc->print_a; \} 58$BR
59 $BBOLD Note:$EBOLD You are allowed only one attempt on this problem.
60 $PAR 61 END_TEXT 62 Context()->normalStrings; 63 64 ANS(str_cmp($mc->correct_ans));
65
66 $j1 = 0; 67 68 Context()->texStrings; 69 SOLUTION(EV3(<<'END_SOLUTION')); 70$PAR
71 $SOL 72$PAR
73 For $$n \ge 1$$, $$0 \le b^{a} n \le 1$$, so
74
75 $0 \le func \le \frac{1}{n^{np}} .$
76
77 The series $$\displaystyle \sum_{n=start}^{\infty} \frac{1}{n^{np}}$$ is a p-series with $$p = np > 1$$, so it converges.  By the Comparison Test we can therefore conclude that series $$\sum\limits_{n=start}^{\infty} func$$ also converges.
78 \$PAR
79 END_SOLUTION
80
81 ENDDOCUMENT();
82
83