View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.2_Summing_an_Infinite_Series/10.2.34.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Limits and Derivatives')
    3 ## DBsection('Definition of the Derivative')
    4 ## KEYWORDS('calculus', 'derivatives', 'slope')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.2')
    9 ## Problem1('34')
   10 ## Author('Keith Thompson')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("Parser.pl");
   16 loadMacros("freemanMacros.pl");
   17 loadMacros("PGauxiliaryFunctions.pl");
   18 loadMacros("PGgraphmacros.pl");
   19 
   20 #$showPartialCorrectAnswers=1;
   21 
   22 $a=random(1,3);
   23 $c=random(3,5);
   24 $sp=random(3,5,1);
   25 $sp1=$sp+1;
   26 $sp2=$sp+2;
   27 $sp3=$sp+3;
   28 $num=2*$a;
   29 $den=2*$c+1;
   30 $ans1=$num**$sp/($den-$num);
   31 Context()->texStrings;
   32 BEGIN_TEXT
   33 \{ beginproblem() \}
   34 \{ textbook_ref_exact("Rogawski ET 2e", "10.2","34") \}
   35 $PAR
   36 
   37 Use the formula for the sum of a geometric series to find the sum or state that the series diverges (enter DIV for a divergent series).
   38 
   39 
   40 \[\frac{$num^{$sp}}{$den}+\frac{$num^{$sp1}}{$den^2}+\frac{$num^{$sp2}}{$den^3}+\frac{$num^{$sp3}}{$den^4}+\cdots\]
   41 
   42 \(S=\) \{ans_rule()\}
   43 END_TEXT
   44 
   45 Context()->normalStrings;
   46 
   47 ANS(std_num_str_cmp($ans1,['DIV'])).
   48 
   49 Context()->texStrings;
   50 SOLUTION(EV3(<<'END_SOLUTION'));
   51 $PAR
   52 $SOL
   53 This is a geometric series with \(c=\frac{$num^{$sp}}{$den}\) and \(0<r=\frac{$num}{$den}<1\). Thus,
   54 \[\sum_{n=0}^\infty \frac{$num^{$sp}}{$den}\left(\frac{$num}{$den}\right)^n=\frac{$num^{$sp}}{$den}\left(\frac{1}{1-\frac{$num}{$den}}\right)=\frac{$num^{$sp}}{$den-$num}.\]
   55 END_SOLUTION
   56 ENDDOCUMENT();