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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2191 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('35')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
15
19
21
22 #Book Values
23 #$constant = -3; 24 25$constant = random(2, 9);
26 $sign = list_random(-1, 1); 27$coef = $constant*$sign;
28 $denominator = Formula("1 -$coef*x")->reduce;
29 # Edit
30 $answer= Compute("($coef*x)^n");
31 $answer->{test_points}=[[1,0.1],[2,0.2],[3,0.3]]; 32 33 #I'm putting in computing the interval of convergence in up here 34 #compute the left endpoint 35$leftendpoint = -1/($constant); 36$rightendpoint= 1/($constant); 37 38 Context()->texStrings; 39 40 BEGIN_TEXT 41 \{ beginproblem() \} 42 \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "35") \} 43$PAR
44 Use Eq. (1) from the text to expand the function into a power series with center $$c= 0$$
45 and determine the set of $$x$$ for which the expansion is valid.
46 $PAR 47 $f(x) = \frac{1}{denominator}$ 48$PAR
49 $$\frac{1}{denominator} = \sum\limits_{n=0}^{\infty}$$ \{ans_rule() \}
50 $PAR 51 The interval of convergence is: \{ans_rule() \} 52 END_TEXT 53 54 Context()->normalStrings; 55 56 #Answer Check Time! 57 #ANS(Formula("($coef*x)^n")->cmp);
58 ANS($answer->cmp); 59 ANS(Interval("($leftendpoint, $rightendpoint)")->cmp); 60 61 Context()->texStrings; 62 SOLUTION(EV3(<<'END_SOLUTION')); 63$PAR
64 \$SOL
65 Substituting $$coef x$$ for $$x$$ in Eq. (1), we obtain
66 $\frac{1}{1-coef x} = \sum_{n=0}^{\infty} (coef x)^n = \sum_{n=0}^{\infty} (coef)^n x^n.$
67 This series is valid for $$|constant x| < 1$$, or $$|x| < \frac{1}{constant}$$.  Thus, the interval of convergence is $$\left(\frac{-1}{constant},\frac{1}{constant} \right)$$.
68
69 END_SOLUTION
70
71 ENDDOCUMENT()