View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.1_Sequences/10.1.23.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Infinite Sequences and Series')
    3 ## DBsection('Infinite Sequences and Series')
    4 ## KEYWORDS('calculus', 'series')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.1')
    9 ## Problem1('23')
   10 ## Author('Danny Glin')
   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 
   18 Context()->texStrings;
   19 Context()->variables->add(n=>'Real');
   20 
   21 #$p=1;
   22 #$q=1;
   23 #$r=1;
   24 $p=random(1,9,1);
   25 $q=random(1,9,1);
   26 $qq=$q**2;
   27 $r=random(1,9,1);
   28 
   29 ##for variation in answer, if $pn is 2 nonzero, if $pn is 3, zero
   30 $pn = random(2,3,1);
   31 
   32 $an=Formula("($p n)/sqrt($qq n^($pn)+$r)")->reduce;
   33 $ax = $an->substitute(n=>x);
   34 if($pn==3){
   35   $ans=0;
   36   Context()->flags->set(reduceConstants=>0);
   37   $ax1=Formula("($p x/x^(3/2))/(sqrt($qq x^($pn)+$r)/(x^(3/2)))");
   38   $ax2=Formula("($p/sqrt(x))/sqrt($qq+$r/x^3)");
   39   $solText = "
   40 We have \(a_n=f(n)\) where \(f(x)= $ax\).  Thus
   41 \[
   42 \lim_{n\to\infty}$an =
   43 \lim_{x\to\infty}$ax =
   44 \lim_{x\to\infty}$ax1
   45 \]
   46 \[
   47  = \lim_{x\to\infty}$ax2 =
   48 \frac{0}{\sqrt{$qq+0}} =
   49 \frac{0}{$q} =
   50 $ans.
   51 \]
   52 "
   53 
   54 
   55 }
   56 else{
   57   $ans = $p/$q;
   58   $ax1=Formula("(($p x)/x)/(sqrt($qq x^($pn)+$r)/x)");
   59   $ax2 = Formula("$p /(sqrt($qq +$r/x^($pn)))");
   60   $solText = "
   61 We have \(a_n=f(n)\) where \(f(x)=$ax\).  Thus
   62 \[
   63 \lim_{n\to\infty}a_n =
   64 \lim_{x\to\infty}f(x) =
   65 \lim_{x\to\infty} $ax
   66 \]
   67 \[
   68 = \lim_{x\to\infty} $ax1 =
   69 \lim_{x\to\infty} $ax2 =
   70 \frac{$p}{$q} = $ans.
   71 \]
   72 
   73 
   74 ";
   75 
   76 }
   77 
   78 
   79 
   80 BEGIN_TEXT
   81 \{ beginproblem() \}
   82 \{ textbook_ref_exact("Rogawski ET 2e", "10.1","23") \}
   83 $PAR
   84 Use Theorem 1 to determine the limit of the sequence or type DIV if the sequence diverges.$PAR
   85 \(a_n=$an\)
   86 $PAR
   87 \(\lim\limits_{n\to\infty}a_n = \)  \{ans_rule()\}$BR
   88 END_TEXT
   89 Context()->normalStrings;
   90 
   91 ANS(std_num_str_cmp($ans,['DIV']));
   92 
   93 Context()->texStrings;
   94 Context()->flags->set(reduceConstants=>0);
   95 
   96 
   97 SOLUTION(EV3(<<'END_SOLUTION'));
   98 $PAR
   99 $SOL
  100 $solText
  101 
  102 END_SOLUTION
  103 
  104 ENDDOCUMENT();