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| 1 : | aubreyja | 2584 | # DBsubject('Calculus') |
| 2 : | jj | 3224 | ## DBchapter('Infinite Sequences and Series') |
| 3 : | jj | 3225 | ## DBsection('Absolute Convergence and the Ratio and Root Tests') |
| 4 : | aubreyja | 2584 | # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test') |
| 5 : | # TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | # EditionText1('2') | ||
| 7 : | # AuthorText1('Rogawski') | ||
| 8 : | # Section1('10.5') | ||
| 9 : | # Problem1('11') | ||
| 10 : | # Author('LA Danielson') | ||
| 11 : | # Institution('The College of Idaho') | ||
| 12 : | DOCUMENT(); | ||
| 13 : | |||
| 14 : | |||
| 15 : | |||
| 16 : | #Load Necessary Macros | ||
| 17 : | |||
| 18 : | loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", ); | ||
| 19 : | loadMacros("Parser.pl"); | ||
| 20 : | loadMacros("freemanMacros.pl"); | ||
| 21 : | |||
| 22 : | |||
| 23 : | Context()->variables->add(n=>'Real'); | ||
| 24 : | |||
| 25 : | $exp = Real(random(30,90,10)); | ||
| 26 : | |||
| 27 : | ($series, $num1, $den1, $num2, $den2, $result, $L, $rho, $compare, $answer, $trueanswer,) = @{list_random( | ||
| 28 : | [ "\frac{e^n}{n!}", "e^{n+1}", "(n+1)!", "n!", "e^n", "\frac{e}{n+1}", 0, Real(0), "<", "converges", 'convergent'], | ||
| 29 : | [ "\frac{e^n}{n^n} ", "e^{n+1}", "(n+1)^{n+1}", "n^n", ,"e^n","\frac{e}{n+1}\left( \frac{n}{n+1}\right)^n = \frac{e}{n+1}\left( 1+\frac{1}{n}\right)^{-n}", "0\cdot \frac{1}{e}=0", Real(0), "<", "converges", 'convergent'], | ||
| 30 : | ["\frac{n^{$exp}}{n!}", "(n+1)^{$exp}", "(n+1)!", "n!", "n^{$exp}", "\frac{1}{n+1}\left( \frac{n+1}{n}\right)^{$exp}=\frac{1}{n+1}\left( 1+\frac{1}{n}\right)^{$exp}", "0\cdot 1=0", Real(0), "<", "converges", 'convergent' ])}; | ||
| 31 : | |||
| 32 : | |||
| 33 : | |||
| 34 : | |||
| 35 : | |||
| 36 : | $question = new_multiple_choice(); | ||
| 37 : | $question->qa(' \( \sum_{n=1}^{\infty} $series \) is:', 'convergent'); | ||
| 38 : | $question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive'); | ||
| 39 : | |||
| 40 : | |||
| 41 : | Context()->texStrings; | ||
| 42 : | |||
| 43 : | BEGIN_TEXT | ||
| 44 : | \{ beginproblem() \} | ||
| 45 : | \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "11") \} | ||
| 46 : | $PAR | ||
| 47 : | Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. | ||
| 48 : | \[ \sum\limits_{n=1}^{\infty} $series \] | ||
| 49 : | \(\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =\) \{ans_rule()\} (Enter 'inf' for \(\infty\).) | ||
| 50 : | $PAR | ||
| 51 : | |||
| 52 : | \{ $question->print_q() \} | ||
| 53 : | \{ $question->print_a() \} | ||
| 54 : | END_TEXT | ||
| 55 : | |||
| 56 : | Context()->normalStrings; | ||
| 57 : | |||
| 58 : | |||
| 59 : | ANS($rho->cmp); | ||
| 60 : | ANS(radio_cmp($question->correct_ans)); | ||
| 61 : | |||
| 62 : | Context()->texStrings; | ||
| 63 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 64 : | $PAR | ||
| 65 : | $SOL | ||
| 66 : | With \( a_n = $series \), | ||
| 67 : | \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{$num1}{$den1} \cdot \frac{$num2}{$den2} = $result \] and \[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = $L $compare 1. \] | ||
| 68 : | Therefore, the series \( \sum\limits_{n=1}^{\infty} $series \) $answer by the Ratio Test. | ||
| 69 : | |||
| 70 : | |||
| 71 : | END_SOLUTION | ||
| 72 : | |||
| 73 : | ENDDOCUMENT() |
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