# DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Absolute Convergence and the Ratio and Root Tests') # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test') # TitleText1('Calculus: Early Transcendentals') # EditionText1('2') # AuthorText1('Rogawski') # Section1('10.5') # Problem1('11') # Author('LA Danielson') # Institution('The College of Idaho') DOCUMENT(); #Load Necessary Macros loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", ); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); Context()->variables->add(n=>'Real'); $exp = Real(random(30,90,10)); ($series, $num1,$den1, $num2,$den2, $result,$L, $rho,$compare, $answer,$trueanswer,) = @{list_random( [ "\frac{e^n}{n!}", "e^{n+1}", "(n+1)!", "n!", "e^n", "\frac{e}{n+1}", 0, Real(0), "<", "converges", 'convergent'], [ "\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'], ["\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)^{$exp}", "0\cdot 1=0", Real(0), "<", "converges", 'convergent' ])};$question = new_multiple_choice(); $question->qa(' $$\sum_{n=1}^{\infty} series$$ is:', 'convergent');$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive'); Context()->texStrings; BEGIN_TEXT \{ beginproblem() \} \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "11") \} $PAR Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. $\sum\limits_{n=1}^{\infty} series$ $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.)$PAR \{ $question->print_q() \} \{$question->print_a() \} END_TEXT Context()->normalStrings; ANS($rho->cmp); ANS(radio_cmp($question->correct_ans)); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION')); $PAR$SOL With $$a_n = series$$, $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num1}{den1} \cdot \frac{num2}{den2} = result$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L compare 1.$ Therefore, the series $$\sum\limits_{n=1}^{\infty} series$$ \$answer by the Ratio Test. END_SOLUTION ENDDOCUMENT()