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Sat Sep 8 05:17:01 2007 UTC (5 years, 9 months ago) by sh002i
File size: 3410 byte(s)
Added tags for Rogawski's "Calculus: Early Transcendentals".


    1 ## DESCRIPTION
2 ## Calculus
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('series','divergent','convergent','absolute convergence','root test','conditionally convergent')
6 ## Tagged by cmd6a 5/6/06
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Infinite Sequences and Series')
10 ## DBsection('Absolute Convergence and the Ratio and Root Tests')
11 ## Date('')
12 ## Author('')
13 ## Institution('Rochester')
14 ## TitleText1('')
15 ## EditionText1('')
16 ## AuthorText1('')
17 ## Section1('')
18 ## Problem1('')
19 ## TitleText2('Calculus: Early Transcendentals')
20 ## EditionText2('1')
21 ## AuthorText2('Rogawski')
22 ## Section2('10.5')
23 ## Problem2('37')
24
25 DOCUMENT();        # This should be the first executable line in the problem.
26
28 "PG.pl",
29 "PGbasicmacros.pl",
30 "PGchoicemacros.pl",
32 "PGauxiliaryFunctions.pl"
33 );
34
35 # No partial credit on this problem, so we say:
37
38
39 TEXT(beginproblem());
40 $showPartialCorrectAnswers = 0; 41 42$a_numb = random(1,5,1);
43 $a_sign = random(-1,1,2); 44$a      = $a_numb *$a_sign;
45 if ($a == 1) {$coef_a = ""};
46 if ($a == -1) {$coef_a = "-"};
47 if ($a_numb > 1) {$coef_a = $a}; 48 49$b_numb = random(1,9,1);
50 $b_sign = random(-1,1,2); 51$b      = $b_numb *$b_sign;
52
53 $c_numb =$a_numb;
54 while ($c_numb ==$a_numb) {$c_numb = random(1,5,1)}; 55$c_sign = random(-1,1,2);
56 $c =$c_numb * $c_sign; 57 if ($c == 1) {$coef_c = ""}; 58 if ($c == -1) {$coef_c = "-"}; 59 if ($c_numb > 1) {$coef_c =$c};
60
61 $d_numb = random(1,9,1); 62$temp   = $a_numb *$b_numb;
63 if ($temp == 1) {$d_sign = -$c_sign } 64 else {$d_sign = $c_sign }; 65$d      = $d_numb *$d_sign;
66
67 $exp_num = random(1,2,1); 68 if ($exp_num == 1) {$coef_en = "n"} 69 else {$coef_en = "$exp_num n"}; 70 71$exp_den = random(1,2,1);
72 if ($exp_den == 1) {$coef_ed = "n"}
73 else {$coef_ed = "$exp_den n"};
74
75 if ($exp_num >$exp_den) {
76   $soln1 = 'INF'; 77$soln2 = 'B';
78   };
79 if ($exp_num <$exp_den) {
80   $soln1 = 0; 81$soln2 = 'A';
82   };
83 if ($exp_num ==$exp_den) {
84   $soln1 = ($a_numb/$c_numb)**$exp_num;
85   if ($soln1 < 1) {$soln2 = 'A'}
86   else {$soln2 = 'B'}; 87 }; 88 89$numerator = "($coef_a n +$b)^{$coef_en}"; 90$denominator = "($coef_c n +$d)^{$coef_ed}"; 91 92$frac = "\frac{ $numerator }{$denominator }";
93
94 TEXT(EV2(<<EOT));
95
96 Consider the series
97 $$\displaystyle \sum_{n=1}^{\infty} a_n$$
98 where
99 $a_n = frac$
100 In this problem you must attempt to use the Root Test to decide
101 whether the series converges.
102
103 $BR$BR
104
105 Compute
106 $L=\lim_{n\rightarrow\infty} \sqrt[n]{|a_n|}$
107 Enter the numerical value of the limit L if it converges,
108 INF if it diverges to infinity, MINF if it diverges to negative infinity,
109 or DIV if it diverges but not to infinity or negative infinity. $BR 110 $$L =$$ \{ans_rule( 30) \} 111 112$BR $BR 113 114 Which of the following statements is true?$BR
115 A. The Root Test says that the series converges absolutely. $BR 116 B. The Root Test says that the series diverges.$BR
117 C. The Root Test says that the series converges conditionally. $BR 118 D. The Root Test is inconclusive, but the series converges 119 absolutely by another test or tests.$BR
120 E. The Root Test is inconclusive, but the series diverges
121   by another test or tests. $BR 122 F. The Root Test is inconclusive, but the series converges 123 conditionally by another test or tests.$BR
124 Enter the letter for your choice here:
125 \{ans_rule( 10) \}
126
127 EOT
128 ANS(num_cmp($soln1, strings=>['INF', 'MINF', 'DIV'])); 129 ANS(str_cmp($soln2));
130
131 ENDDOCUMENT();        # This should be the last executable line in the problem.