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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 3005 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('39')
10 # Author('Emily Price')
11 # Institution('W.H.Freeman')
12 DOCUMENT();
13
15
19
21
22 #Book Values
23 #$exponent = 9; 24 #$coeff = 9;
25
26 $exponent = random(2,9); 27$coeff = random(2,9,1); #added LAD for greater variation of answers
28
29 #I'm putting in computing the interval of convergence in up here
30 #compute the left endpoint
31 $leftendpoint = -$coeff**(1/$exponent); 32$rightendpoint= $coeff**(1/$exponent);
33
34 #Context()->variables->set(x=>{limits=>[$leftendpoint,$rightendpoint]});
35 #Context()->variables->set(x=>{limits=>[0.1,$rightendpoint]}); 36 37 #Context()->variables->set(n=>{limits=>[0.1,1]}); 38 39 #$pn = Compute("(-x)^{$exponent*n}/($coeff^{n+1})");
40 $pn = Compute("((-1)^{n})*(x)^{$exponent*n}/($coeff^{n+1})"); 41$pn->{test_points}=[[1,0.1],[2,0.2],[3,0.3]];
42 $point1=.25; 43$point2=.5;
44 $point3=1; 45 #$pn->{test_points} = [[$point1,$leftendpoint],[$point2,0],[$point3,$rightendpoint]]; 46 47 Context()->texStrings; 48 49 BEGIN_TEXT 50 \{ beginproblem() \} 51 \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "39") \} 52$PAR
53 Use Eq. (1) from the text to expand the function into a power series with center $$c= 0$$
54 and determine the set of $$x$$ for which the expansion is valid.
55 $PAR 56 $f(x) = \frac{1}{coeff+x^{exponent}}$ 57$PAR
58 $$\frac{1}{coeff+x^{exponent}} = \sum\limits_{n=0}^{\infty}$$ \{ans_rule() \}
59 $PAR 60 The interval of convergence is: \{ans_rule() \} 61 END_TEXT 62 63 Context()->normalStrings; 64 65 #Answer Check Time! 66 ANS($pn->cmp);
67 ANS(Interval("($leftendpoint,$rightendpoint)")->cmp);
68
69 Context()->texStrings;
70 SOLUTION(EV3(<<'END_SOLUTION'));
71 $PAR 72$SOL
73 First write
74 $\frac{1}{coeff+x^{exponent}} = \frac{1}{coeff}\cdot\frac{1}{1--\frac{x^{exponent}}{coeff}}.$
75 Substituting $$\frac{-x^{exponent}}{coeff}$$ for $$x$$ in Eq. (1), we obtain
76 $\frac{1}{1+\frac{x^{exponent}}{coeff}} 77 = \sum_{n=0}^{\infty} \left(\frac{-x^{exponent}}{coeff}\right)^n 78 = \sum_{n=0}^{\infty} (-1)^n \frac{x^{exponent n}}{coeff^n}.$
79 Thus,
80 $\frac{1}{coeff+x^{exponent}}=\frac{1}{coeff}\sum_{n=0}^{\infty}(-1)^n \frac{x^{exponent n}}{coeff^n} 81 = \sum_{n=0}^{\infty} (-1)^n \frac{x^{exponent n}}{coeff^{n+1}}.$
82 This series is valid for $$\left|\frac{-x^{exponent}}{coeff}\right| < 1$$,
83 or $$|x| < \sqrt[exponent]{coeff}$$. Thus, the interval of convergence is
84 $$\left( -\sqrt[exponent]{coeff}, \sqrt[exponent]{coeff} \right).$$
85
86 END_SOLUTION
87
88 ENDDOCUMENT()