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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('15')
   10 # Author('Emily Price')
   11 # Institution('W.H.Freeman')
   12 DOCUMENT();
   13 
   14 
   15 
   16 #Load Necessary Macros
   17 
   18 loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", );
   19 loadMacros("Parser.pl");
   20 loadMacros("freemanMacros.pl");
   21 
   22 
   23 Context()->variables->add(n=>'Real');
   24 
   25 #Book Values
   26 #$exponent = 2;
   27 
   28 $exponent = random(2, 9);
   29 
   30 Context()->texStrings;
   31 
   32 BEGIN_TEXT
   33 \{ beginproblem() \}
   34 \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "15") \}
   35 $PAR
   36 Find the interval of convergence for the following power series:
   37 $PAR
   38 \[ \sum_{n=1}^{\infty} \frac{x^n}{(n!)^{$exponent}} \]
   39 $PAR
   40 The interval of convergence is: \{ans_rule() \}
   41 END_TEXT
   42 
   43 
   44 Context()->normalStrings;
   45 
   46 #Answer Check Time!
   47 ANS(Interval("(-inf, inf)")->cmp);
   48 
   49 Context()->texStrings;
   50 SOLUTION(EV3(<<'END_SOLUTION'));
   51 $PAR
   52 $SOL
   53 With \( a_n = \frac{1}{(n!)^{$exponent}}\),
   54 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{((n+1)!)^{$exponent}} \cdot \frac{(n!)^{$exponent}}{1} = \left( \frac{1}{n+1} \right)^{$exponent} \]
   55 and
   56 \[ r = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0. \]
   57 The radius of convergence is therefore \(R = r^{-1} = \infty\), and the series converges absolutely for all \(x\).  Thus, the interval of convergence is \((-\infty,\infty) \).
   58 
   59 END_SOLUTION
   60 
   61 ENDDOCUMENT()