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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2587 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('51')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
18 $context = Context(); 19 20$context->variables->add(n=>'Real');
21
22 $a = Real(random(3, 7, 1)); 23$b = Real(random(6, 9, 1));
24 $p = Real(random(3, 7, 1)); 25$pm1 = $p-1; 26$p2m1 = 2*$p-1; 27 28$start = 1;
29
30
31
32 $ch = random(0,1,1); ##0 diverge, 1 converge! LAD 33 34 if ($ch >0) {#converge
35     $answer = "converges"; 36$type = "convergent";
37     $wrong = "diverges"; 38$start = 1;
39     $func = "\frac{\sqrt[$p]{n}}{$a n^2 +$b}";
40     $bn = "\frac{1}{n^{\frac{$p2m1}{$p}}}"; 41$cn = "\frac{n^2}{$a n^2 +$b}";
42     $pnum =$p2m1;
43     $compare = ">"; 44 45 }else{ 46$answer = "diverges";
47     $type = "divergent"; 48$wrong = "converges";
49     $start = 1; 50$func = "\frac{\sqrt[$p]{n}}{$a n + $b}"; 51$bn = "\frac{1}{n^{\frac{$pm1}{$p}}}";
52     $cn = "\frac{n}{$a n + $b}"; 53$pnum = $pm1; 54$compare = "<";
55 }
56
57
58 $mc = new_multiple_choice(); 59 60$mc->qa("Determine convergence or divergence of $$\displaystyle \sum_{n=start}^{\infty} func$$.",
61     $answer); 62$mc->extra($wrong); 63$mc->makeLast("diverges");
64
65 Context()->texStrings;
66 BEGIN_TEXT
67 \{ beginproblem() \}
68 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","51") \}
69 $PAR 70 \{$mc->print_q; \}
71 \{ $mc->print_a; \} 72$BR
73 $BBOLD Note:$EBOLD You are allowed only one attempt on this problem.
74 $PAR 75 END_TEXT 76 Context()->normalStrings; 77 78 ANS(str_cmp($mc->correct_ans));
79
80 $j1 = 0; 81 82 Context()->texStrings; 83 SOLUTION(EV3(<<'END_SOLUTION')); 84$PAR
85 $SOL 86$PAR
87 Apply the Limit Comparison Test with $$a_n = func$$ and $$b_n = bn$$:
88
89 $L = \lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} \frac{func}{bn} = \lim_{n\to\infty} cn = \frac{1}{a} .$
90
91 The series $$\displaystyle \sum_{n=1}^{\infty} bn$$ is a $type $$p$$-series $$\left( p = \frac{pnum}{p} compare 1\right)$$. Because $$L > 0$$ exists, by the Limit Comparison Test we can conclude that the series $$\displaystyle \sum_{n=1}^{\infty} func$$ also$answer.
92 \$PAR
93 END_SOLUTION
94
95 ENDDOCUMENT();
96
97