[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 10_Infinite_Series / 10.5_The_Ratio_and_Root_Tests / 10.5.56.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.56.pg

Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 1908 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

1 # DBsubject('Calculus')
2 # DBchapter('Infinite Series and Sequences')
3 # DBsection('Absolute Convergence and the Root and Ratio Tests')
4 # KEYWORDS('calculus', 'series', 'sequences', 'convergence', 'root test', 'ratio test')
5 # TitleText1('Calculus: Early Transcendentals')
6 # EditionText1('2')
7 # AuthorText1('Rogawski')
8 # Section1('10.5')
9 # Problem1('56')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
16 #Load Necessary Macros
17
21
22
24
25 #Book Values
26 #$constant = 12 27 28$constant = random(5, 35);
29
30
31 #Let's try to make a multiple choice question
32 $question = new_multiple_choice(); 33$question->qa(' $$\sum\limits_{n=1}^{\infty} ( \frac{n}{n+constant} )^n$$ is:', 'divergent');
34 $question->makeLast( 'convergent', 'divergent'); 35 36 37 Context()->texStrings; 38 39 BEGIN_TEXT 40 \{ beginproblem() \} 41 \{ textbook_ref_exact("Rogawski ET 2e", "10.5", "56") \} 42$PAR
43 Determine convergence or divergence using any method covered so far.
44 $PAR 45 \{$question->print_q() \}
46 \{ $question->print_a() \} 47 END_TEXT 48 49 Context()->normalStrings; 50 51 #Answer Check Time! 52 ANS(radio_cmp($question->correct_ans));
53
54 Context()->texStrings;
55 SOLUTION(EV3(<<'END_SOLUTION'));
56 $PAR 57$SOL
58 Because the general term has the form of a function of $$n$$ raised to the $$n$$th power, we might be tempted to use the Root Test; however, the Root Test is inconclusive for this series.  Instead, note
59 $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 1 + \frac{constant}{n} \right)^{-n} = \lim_{n \to \infty} \left[ \left( 1 + \frac{constant}{n} \right)^{\frac{n}{constant}} \right]^{-constant} = e^{-constant} \neq 0.$
60 Therefore, the series diverges by the Divergence Test.
61
62
63 END_SOLUTION
64
65 ENDDOCUMENT()

 aubreyja at gmail dot com ViewVC Help Powered by ViewVC 1.0.9