View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.6_Power_Series/10.6.1.pg

    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('1')
   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 #$base = 2;
   27 
   28 $base = random(2, 29);
   29 $answer = Formula("$base");
   30 
   31 Context()->texStrings;
   32 
   33 BEGIN_TEXT
   34 \{ beginproblem() \}
   35 \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "1") \}
   36 $PAR
   37 Use the Ratio Test to determine the radius of convergence of the following series:
   38 $PAR
   39 \[ \sum_{n=0}^{\infty} \frac{x^n}{$base^n} \]
   40 $PAR
   41 \( R = \) \{ans_rule() \}
   42 END_TEXT
   43 
   44 
   45 Context()->normalStrings;
   46 
   47 #Answer Check Time!
   48 ANS($answer->cmp);
   49 
   50 Context()->texStrings;
   51 SOLUTION(EV3(<<'END_SOLUTION'));
   52 $PAR
   53 $SOL
   54 
   55 With \( a_n = \frac{x^n}{$base^n} \),
   56 \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|^{n+1}}{$base^{n+1}} \cdot \frac{$base^n}{|x|^n} = \frac{|x|}{$base} \]
   57 and
   58 \[\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|}{$base}. \]
   59 By the Ratio Test, the series converges when \( \rho = \frac{|x|}{$base} < 1\), or \(|x| < $base \), and diverges when \(\rho = \frac{|x|}{$base} > 1\), or \(|x| > $base\).  The radius of convergence is therefore \( R = $base \).
   60 
   61 END_SOLUTION
   62 
   63 ENDDOCUMENT()