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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2930 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('7')
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
24 $a = Real(random(1, 9, 1)); 25$start = $a; 26$asqr = $a**2; 27$a4 = 4*$a; 28$spasqr = $start +$asqr;
29
30 $int_lim = Real(pi/(4*$a));
31
32 ($func,$f, $ans_lim,$func2, $lim,$answer) = @{ list_random(
33 ["\frac{1}{n^2 + $asqr}", 34 Formula("1/(x^2+$asqr)"),
35 $int_lim, 36 "\frac{1}{x^2 +$asqr}",
37 "\frac{1}{$a}\lim_{R\to\infty}\left(\tan^{-1}\left(\frac{R}{$a}\right) - \tan^{-1}\left(\frac{$a}{$a}\right)\right) = \frac{1}{$a}\left(\frac{\pi}{2} - \frac{\pi}{4}\right) = \frac{\pi}{$a4}",
38 "converges"],
39 ["\frac{1}{n + $asqr}", 40 Formula("1/(x+$asqr)"),
41 "inf",
42 "\frac{1}{x + $asqr}", 43 "\lim_{R\to\infty}\left(\ln(R+$asqr) - \ln($spasqr)\right) = \infty", 44 "diverges"] 45 ) }; 46 47 48 49$wrong = "converges";
50
51 if ($answer eq "converges") { 52$wrong = "diverges";
53 }
54
55 $mc = new_multiple_choice(); 56 57$mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ",
58     $answer); 59$mc->extra($wrong); 60$mc->makeLast("diverges");
61
62 Context()->texStrings;
63 BEGIN_TEXT
64 \{ beginproblem() \}
65 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","7") \}
66 $PAR 67 Use the Integral Test to determine whether the infinite series is convergent. 68 $\sum_{n=start}^{\infty} func$ 69 Fill in the corresponding integrand and 70 the value of the improper integral. 71$BR
72 Enter $BBOLD inf$EBOLD for $$\infty$$, $BBOLD -inf$EBOLD for $$-\infty$$,
73 and $BBOLD DNE$EBOLD if the limit does not exist.
74 $PAR 75 Compare with 76 $$\int_{start}^{\infty}$$ \{ ans_rule() \} $$dx$$ = \{ ans_rule() \} 77$PAR
78 By the Integral Test,
79 $BR 80 \{$mc->print_q; \}
81 \{ $mc->print_a; \} 82$PAR
83 END_TEXT
84 Context()->normalStrings;
85
86 ANS($f->cmp); 87 ANS( num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"]));
88 ANS(str_cmp($mc->correct_ans)); 89 90$j1 = 0;
91
92 Context()->texStrings;
93 SOLUTION(EV3(<<'END_SOLUTION'));
94 $PAR 95$SOL
96 $PAR 97 Let $$f(x) = func2$$. This function is continuous, positive and decreasing on the interval $$x \ge start$$, so the Integral Test applies. Moreover, 98 99 $\int_{start}^{\infty} func2 \, dx = \lim_{R\to\infty} \int_{start}^{R} func2 \, dx$ 100 $= lim$. 101$PAR
102 The integral $answer; hence the series $$\displaystyle \sum_{n=start}^{\infty} func$$ also$answer.
103 \$PAR
104 END_SOLUTION
105
106 ENDDOCUMENT();
107
108