View of /trunk/NationalProblemLibrary/Rochester/setSeries5IntegralTest/ns8_3_24eva.pg

    1 ## DESCRIPTION
    2 ##   Infinite Series with the Integral Test
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('Series', 'Converge', 'Diverge', Integral Test', 'Partial Sum', 'Estimate', 'Error')
    6 ## Tagged by nhamblet
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Infinite Sequences and Series')
   10 ## DBsection('The Integral Test and Estimates of Sums')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGauxiliaryFunctions.pl"
   28 );
   29 
   30 TEXT(beginproblem());
   31 $showPartialCorrectAnswers = 1;
   32 
   33 $a = random(1,10,1);
   34 $b = random(1,10,1);
   35 $c = random(3,8,1);
   36 #$accuracy = random(0,.000001,.00000001);
   37 $d = random(3,6,1);
   38 $c1 = $c - 1 ;
   39 
   40 if ($d==3) {
   41   $nth = '3rd';
   42 } elsif ($d==4) {
   43   $nth = '4th';
   44 } elsif ($d==5) {
   45   $nth = '5th';
   46 } else {
   47   $nth = '6th';
   48 }
   49 
   50 
   51 BEGIN_TEXT
   52 
   53 \( (a) \) Compute \(s_{$d} \) (the $nth partial sum) of
   54   \( \displaystyle s=\sum_{n=1}^{\infty}\frac{$a}{$b n^$c}  \)$BR
   55   \{ans_rule(45)\} $BR
   56 \( (b) \) Estimate the error in using \(s_{$d}\) as an approximation of the sum
   57   of the series. (i.e. use \( \displaystyle \int_{$d}^{\infty} f(x)dx \geq R_{$d} \) ) $BR
   58   \{ans_rule(30)\} $BR
   59 \( (c) \) Use n = $d and
   60 \[ s_n + \int_{n+1}^{\infty} f(x)dx \leq s \leq s_n + \int_{n}^{\infty}f(x)dx \]
   61   to find a better estimate of the sum. $BR
   62   \{ans_rule(35)\} \( \leq s \leq \)  \{ans_rule(35)\}
   63 END_TEXT
   64 
   65 # a simple code to perform a summation. (for A)
   66   $i = 1;
   67   $sum = 0;
   68   while ($i <= $d ) {
   69                 $fofi = $a/($b*$i**$c);
   70     $sum = $sum + $fofi;
   71     $i = $i +1 ;
   72   }
   73 #to make sure that the answer is always an integer (for D)
   74 # $terms =  (1/(3*$accuracy))**(1/3);
   75 # if ($terms != int($terms)) {
   76 #   $terms = $terms +1 ;
   77 #   }
   78 #@ans=();
   79 $ans0 = $sum ;
   80 $ans1 = $a/($b*$c1*$d**$c1);
   81 $ans2 = $sum + $a/($b*$c1*($d+1)**$c1);
   82 $ans3 = $sum + $a/($b*$c1*$d**$c1);
   83 
   84 #ANS(num_cmp(relTol=>@ans));
   85 #ANS(ordered_num_cmp_list(@ans) );
   86 ANS(num_cmp($ans0));
   87 ANS(num_cmp($ans1));
   88 ANS(num_cmp($ans2));
   89 ANS(num_cmp($ans3));
   90 
   91 ENDDOCUMENT();        # This should be the last executable line in the problem.