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    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 
   14 #Load Necessary Macros
   15 
   16 loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", );
   17 loadMacros("Parser.pl");
   18 loadMacros("freemanMacros.pl");
   19 
   20 Context()->variables->add(n=>'Real');
   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()