View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.2_Summing_an_Infinite_Series/10.2.11.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('11')
   10 ## Author('LA Danielson')
   11 ## Institution('The College of Idaho')
   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 $n0 = Real(random(2,9,1));
   23 
   24 sub partial_sum {
   25     $n = shift; #number of terms
   26     $sum = 0;
   27 
   28     for($i=$n0; $i<($n0+$n); $i++){
   29       $sum += (1/($i+1))-(1/($i+2));
   30 
   31     }
   32 
   33     return $sum;
   34 }
   35 
   36 
   37 $ans1=partial_sum(3);
   38 $ans2=partial_sum(4);
   39 $ans3=partial_sum(5);
   40 $ans4=1/($n0+1);
   41 #for solution formatting
   42 $ns[0] = $n0;
   43 for($i=1; $i<=6; $i++){
   44    $ns[$i]= $ns[$i-1]+1;
   45 }
   46 
   47 $num1=$ns[4]-$ns[1];
   48 $den1=$ns[4]*$ns[1];
   49 
   50 $num2=$ns[5]-$ns[1];
   51 $den2=$ns[5]*$ns[1];
   52 
   53 $num3=$ns[6]-$ns[1];
   54 $den3=$ns[6]*$ns[1];
   55 
   56 Context()->texStrings;
   57 BEGIN_TEXT
   58 \{ beginproblem() \}
   59 \{ textbook_ref_exact("Rogawski ET 2e", "10.2","11") \}
   60 $PAR
   61 
   62 Calculate \(S_3,S_{4}\), and \(S_{5}\) and then find the sum for the telescoping series
   63 \[S=\sum_{n=$n0}^\infty \left(\frac{1}{n+1}-\frac{1}{n+2}\right)\]
   64 
   65 $PAR \(S_3\) =  \{ans_rule()\}
   66 $PAR \(S_{4}\) =  \{ans_rule()\}
   67 $PAR \(S_{5}\) =  \{ans_rule()\}
   68 $PAR \(S\) =  \{ans_rule()\}
   69 END_TEXT
   70 
   71 Context()->normalStrings;
   72 
   73 ANS(Real($ans1)->cmp);
   74 ANS(Real($ans2)->cmp);
   75 ANS(Real($ans3)->cmp);
   76 ANS(Real($ans4)->cmp);
   77 
   78 Context()->texStrings;
   79 SOLUTION(EV3(<<'END_SOLUTION'));
   80 $PAR
   81 $SOL
   82 \[S_3=\left(\frac{1}{$ns[1]}-\frac{1}{$ns[2]}\right)+\left(\frac{1}{$ns[2]}-\frac{1}{$ns[3]}\right)+\left(\frac{1}{$ns[3]}-\frac{1}{$ns[4]}\right)=\frac{1}{$ns[1]}-\frac{1}{$ns[4]}=\frac{$num1}{$den1};\]
   83 
   84 \[S_4=S_3+\left(\frac{1}{$ns[4]}-\frac{1}{$ns[5]}\right)=\frac{1}{$ns[1]}-\frac{1}{$ns[5]}=\frac{$num2}{$den2};\]
   85 
   86 \[S_5=S_4+\left(\frac{1}{$ns[5]}-\frac{1}{$ns[6]}\right)=\frac{1}{$ns[1]}-\frac{1}{$ns[6]}=\frac{$num3}{$den3}.\]
   87 
   88 The general term in the sequence of partial sums is
   89 \[S_N=\left(\frac{1}{$ns[1]}-\frac{1}{$ns[2]}\right)+\left(\frac{1}{$ns[2]}-\frac{1}{$ns[3]}\right)+\left(\frac{1}{$ns[3]}-\frac{1}{$ns[4]}\right)+\cdots +\left(\frac{1}{N+$ns[0]}-\frac{1}{N+$ns[1]}\right)=\frac{1}{$ns[1]}-\frac{1}{N+$ns[1]};\]
   90 thus,
   91 \[S=\lim_{N\rightarrow \infty} S_N=\lim_{N\rightarrow \infty}\left(\frac{1}{$ns[1]}-\frac{1}{N+$ns[1]}\right)=\frac{1}{$ns[1]}.\]
   92 $PAR Thus the sum of the telescoping series is \(\frac{1}{$ns[1]}\).
   93 END_SOLUTION
   94 
   95 ENDDOCUMENT();