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Tue Nov 8 15:17:41 2011 UTC (19 months, 1 week ago) by aubreyja
File size: 3035 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('3')
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 $a = Real(random(2, 9, 1)); 24$a2 = $a - 1; 25$a3 = Real(1/$a2); 26$b = Real(random(2,7,1));
27 $bm1 =$b-1;
28 $bp1 =$b+1;
29
30 ($func,$f, $ans_lim,$func2, $lim,$answer) = @{ list_random(
31 ["n^{-\frac{1}{$b}}", Formula(" x^(-1/$b)") , "inf"  , "x^{-\frac{1}{$b}}", "$$\frac{b}{bm1} \displaystyle \lim_{R\to\infty}\left(R^{\frac{bm1}{b}} - 1\right) = \infty$$", "diverges"], 32 ["n^{-\frac{$b}{$bp1}}", Formula("x^(-$b/$bp1) "), "inf","x^{-\frac{$b}{$bp1}}", "$$bp1 \displaystyle \lim_{R\to\infty}\left(R^{\frac{1}{bp1}} - 1\right) = \infty$$", "diverges"], 33 ["n^{-$a}", Formula(" x^(-$a)"),$a3 ,"x^{-$a}", "$$-\frac{1}{a2} \displaystyle \lim_{R\to\infty}\left(R^{-a2} - 1\right) = \frac{1}{a2}$$", "converges"] 34 ) }; 35 36$start = 1;
37
38 $wrong = "converges"; 39 40 if ($answer eq "converges") {
41     $wrong = "diverges"; 42 } 43 44$mc = new_multiple_choice();
45
46 $mc->qa("the infinite series $$\displaystyle \sum_{n=start}^{\infty} func$$ ", 47$answer);
48 $mc->extra($wrong);
49 $mc->makeLast("diverges"); 50 51 Context()->texStrings; 52 BEGIN_TEXT 53 \{ beginproblem() \} 54 \{ textbook_ref_exact("Rogawski ET 2e", "10.3","3") \} 55$PAR
56 Use the Integral Test to determine whether the infinite series is convergent.
57 $\sum_{n=start}^{\infty} func$
58 Fill in the corresponding integrand and
59 the value of the improper integral.
60 $BR 61 Enter$BBOLD inf $EBOLD for $$\infty$$,$BBOLD -inf $EBOLD for $$-\infty$$, 62 and$BBOLD DNE $EBOLD if the limit does not exist. 63$PAR
64 Compare with
65 $$\int_{start}^{\infty}$$ \{ ans_rule() \} $$dx$$ = \{ ans_rule() \}
66 $PAR 67 By the Integral Test, 68$BR
69
70 \{ $mc->print_q; \} 71 \{$mc->print_a; \}
72 $PAR 73 END_TEXT 74 Context()->normalStrings; 75 76 77 ANS($f->cmp);
78 ANS( num_cmp($ans_lim,strings=>["inf","INF", "-inf","-INF","DNE","dne"])); 79 ANS(str_cmp($mc->correct_ans));
80
81 $j1 = 0; 82 83 Context()->texStrings; 84 SOLUTION(EV3(<<'END_SOLUTION')); 85$PAR
86 $SOL 87$PAR
88 Let $$f(x) = func2$$.  This function is continuous, positive and decreasing on the interval $$x \ge start$$, so the Integral Test applies.  Moreover,
89 $PAR 90 $$\displaystyle \int_{start}^{\infty} func2 \, dx = \displaystyle \lim_{R\to\infty} \displaystyle \int_{start}^{R} func2 \, dx =$$$lim.
91 $PAR 92 The integral$answer; hence the series $$\displaystyle \sum_{n=start}^{\infty} func$$ also $answer. 93$PAR
94 END_SOLUTION
95
96 ENDDOCUMENT();
97
98