Parent Directory
| 
Revision Log
Revision 2584 - (view) (download)
| 1 : | aubreyja | 2584 | ## 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(); | ||
| 14 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 15 : | loadMacros("PGchoicemacros.pl"); | ||
| 16 : | loadMacros("Parser.pl"); | ||
| 17 : | loadMacros("freemanMacros.pl"); | ||
| 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 : |
| aubreyja at gmail dot com | ViewVC Help |
| Powered by ViewVC 1.0.9 |