# DBsubject('Calculus') # DBchapter('Infinite Series and Sequences') # DBsection('Absolute Convergence and the Root and Ratio Tests') # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'ratio test') # TitleText1('Calculus: Early Transcendentals') # EditionText1('2') # AuthorText1('Rogawski') # Section1('10.5') # Problem1('5') # Author('Emily Price') # Institution('W.H.Freeman') DOCUMENT(); #Load Necessary Macros loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", ); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); Context()->variables->add(n=>'Real'); #Book Values # $exp1 = 2 #$exp2 = 1 # $constant = 1$exp1 = random(2, 7); $exp2 = random(1,$exp1-1,1);#Formula("$exp1 - 1")->reduce;$constant = random(1, 9); $num = Formula("n^{$exp2}")->reduce; $num2 = Formula("(n+1)^{$exp2}")->reduce; $denominator = "n^{$exp1} + $constant";$denomplus1 = "(n+1)^{$exp1} +$constant"; $rho = Real(1); #Let's try to make a multiple choice question$question = new_multiple_choice(); $question->qa(' $$\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$$ is:', 'The Ratio Test is inconclusive');$question->makeLast( 'convergent', 'divergent', 'The Ratio Test is inconclusive'); Context()->texStrings; BEGIN_TEXT \{ beginproblem() \} \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "5") \} $PAR Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. $\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$ $$\rho = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| =$$ \{ans_rule()\} (Enter 'inf' for $$\infty$$.)$PAR \{ $question->print_q() \} \{$question->print_a() \} END_TEXT Context()->normalStrings; #Answer Check Time! ANS($rho->cmp); ANS(radio_cmp($question->correct_ans)); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION')); $PAR$SOL With $$a_n = \frac{num}{denominator}$$, $\left| \frac{a_{n+1}}{a_n} \right| = \frac{num2}{denomplus1} \cdot \frac{denominator}{num} = \frac{num2}{num} \cdot \frac{denominator}{denomplus1},$ and $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \cdot 1 = 1.$ Therefore, for the series $$\sum\limits_{n=1}^{\infty} \frac{num}{denominator}$$, the Ratio Test is inconclusive. END_SOLUTION ENDDOCUMENT()