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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('35')
   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 #$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()