#DESCRIPTION ## Calculus: Absolute convergence and the ratio and root tests ##ENDDESCRIPTION ##KEYWORDS('calculus', 'series', 'ratio test') ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Absolute Convergence and the Ratio and Root Tests') ## Date('04/07/2010') ## Author('Justin Sukiennik') ## Institution('University of Minnesota') ## TitleText1('Calculus: Concepts and Contexts') ## EditionText1('4 Custom UMTYMP Ed.') ## AuthorText1('Stewart') ## Section1('11.6') ## Problem1('1') ##################################################################### DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGstandard.pl", "MathObjects.pl", "PGunion.pl", "parserPopUp.pl", "PGchoicemacros.pl", "PGcourse.pl", ); ##################################################################### install_problem_grader(~~&std_problem_grader); TEXT(beginproblem()); $showPartialCorrectAnswers = 0; ##################################################################### Context("Numeric");$a = list_random(2,4,5,10,20); $b = Compute("1/$a"); $popup1 = PopUp(['?','Convergent', 'Divergent','Inconclusive'],'Convergent');$popup2 = PopUp(['?','Convergent', 'Divergent','Inconclusive'],'Inconclusive'); $popup3 = PopUp(['?','Convergent', 'Divergent','Inconclusive'],'Divergent'); ##################################################################### #Title("$BITALIC Look Ahead: $EITALIC 11.6 Absolute Convergence and the Ratio and Root Tests"); Context()->texStrings; BEGIN_TEXT What can you say about the series $$\sum a_n$$ in each of the following cases using the Ratio Test? Answer "Convergent," "Divergent," or "Inconclusive."$BR $HR \{$popup1->menu\} $BBOLD 1.$EBOLD $$\displaystyle \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n} \right| = b$$ $PAR \{$popup2->menu\} $BBOLD 2.$EBOLD $$\displaystyle \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n} \right| = 1$$ $PAR \{$popup3->menu\} $BBOLD 3.$EBOLD $$\displaystyle \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n} \right| = a$$ $HR$BBOLD Note:$EBOLD$BITALIC You only have two attempts at this problem.$EITALIC$HR END_TEXT ##################################################################### ANS($popup1->cmp); ANS($popup2->cmp); ANS(\$popup3->cmp); ENDDOCUMENT(); # This should be the last executable line in the problem.