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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 1736 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('Representations of Functions as Power Series')
4 # KEYWORDS('calculus', 'series', 'sequences', 'power series', 'convergence', 'radius of convergence', 'interval of convergence')
5 # TitleText1('Calculus: Early Transcendentals')
6 # EditionText1('2')
7 # AuthorText1('Rogawski')
8 # Section1('10.6')
9 # Problem1('1')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
14
15
17
21
22
24
25 #Book Values
26 #$base = 2; 27 28$base = random(2, 29);
29 $answer = Formula("$base");
30
31 Context()->texStrings;
32
33 BEGIN_TEXT
34 \{ beginproblem() \}
35 \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "1") \}
36 $PAR 37 Use the Ratio Test to determine the radius of convergence of the following series: 38$PAR
39 $\sum_{n=0}^{\infty} \frac{x^n}{base^n}$
40 $PAR 41 $$R =$$ \{ans_rule() \} 42 END_TEXT 43 44 45 Context()->normalStrings; 46 47 #Answer Check Time! 48 ANS($answer->cmp);
49
50 Context()->texStrings;
51 SOLUTION(EV3(<<'END_SOLUTION'));
52 $PAR 53$SOL
54
55 With $$a_n = \frac{x^n}{base^n}$$,
56 $\left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|^{n+1}}{base^{n+1}} \cdot \frac{base^n}{|x|^n} = \frac{|x|}{base}$
57 and
58 $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|}{base}.$
59 By the Ratio Test, the series converges when $$\rho = \frac{|x|}{base} < 1$$, or $$|x| < base$$, and diverges when $$\rho = \frac{|x|}{base} > 1$$, or $$|x| > base$$.  The radius of convergence is therefore $$R = base$$.
60
61 END_SOLUTION
62
63 ENDDOCUMENT()