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Tue Nov 8 15:17:41 2011 UTC (18 months, 1 week ago) by aubreyja
File size: 2084 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('19')
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 $a = Real(random(3, 9, 1)); 24 25$start = 1;
26
27 $answer = "converges"; 28 29$func = "\frac{1}{n$a^n}"; 30 31$wrong = "converges";
32
33 if ($answer eq "converges") { 34$wrong = "diverges";
35 }
36
37 $mc = new_multiple_choice(); 38 39$mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ",
40     $answer); 41$mc->extra($wrong); 42$mc->makeLast("diverges");
43
44 Context()->texStrings;
45 BEGIN_TEXT
46 \{ beginproblem() \}
47 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","19") \}
48 $PAR 49 Use the Comparison Test to determine whether the infinite series is convergent. 50 $\sum_{n=start}^{\infty} func$ 51 52 By the Comparison Test, 53 \{$mc->print_q; \}
54 \{ $mc->print_a; \} 55$BR
56 $BBOLD Note:$EBOLD You are allowed only one attempt on this problem.
57 $PAR 58 END_TEXT 59 Context()->normalStrings; 60 61 ANS(str_cmp($mc->correct_ans));
62
63 $j1 = 0; 64 65 Context()->texStrings; 66 SOLUTION(EV3(<<'END_SOLUTION')); 67$PAR
68 $SOL 69$PAR
70 We compare with the geometric series $$\displaystyle \sum \left(\frac{1}{a}\right)^n$$.  For $$n \ge 1$$,
71
72 $func \le \frac{1}{a^n} = \left(\frac{1}{a}\right)^n .$
73
74 Since $$\displaystyle \sum_{n=start}^{\infty} \left(\frac{1}{a}\right)^n$$ converges (it's a geometric series with $$|r| = \frac{1}{a} <1$$ ), we conclude by the Comparison Test that $$\displaystyle \sum_{n=start}^{\infty} func$$ also converges.
75 \$PAR
76 END_SOLUTION
77
78 ENDDOCUMENT();
79
80


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