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| 1 : | glarose | 964 | # DESCRIPTION |
| 2 : | # Problem from Calculus, single variable, Hughes-Hallett et al., 4th ed. | ||
| 3 : | # WeBWorK problem written by Gavin LaRose, <glarose@umich.edu> | ||
| 4 : | # ENDDESCRIPTION | ||
| 5 : | |||
| 6 : | ## KEYWORDS('calculus', 'integral', 'series', 'power series', 'interval of convergence', 'radius of convergence') | ||
| 7 : | ## Tagged by glr 02/08/09 | ||
| 8 : | |||
| 9 : | ## DBsubject('Calculus') | ||
| 10 : | ## DBchapter('Infinite Sequences and Series') | ||
| 11 : | ## DBsection('Power Series') | ||
| 12 : | ## Date('') | ||
| 13 : | ## Author('Gavin LaRose') | ||
| 14 : | ## Institution('University of Michigan') | ||
| 15 : | ## TitleText1('Calculus') | ||
| 16 : | ## EditionText1('4') | ||
| 17 : | ## AuthorText1('Hughes-Hallett') | ||
| 18 : | ## Section1('9.5') | ||
| 19 : | ## Problem1('15') | ||
| 20 : | |||
| 21 : | ## Textbook tags | ||
| 22 : | ## HHChapter('Sequences and Series') | ||
| 23 : | ## HHSection('Power Series and Interval of Convergence') | ||
| 24 : | |||
| 25 : | |||
| 26 : | |||
| 27 : | DOCUMENT(); | ||
| 28 : | |||
| 29 : | loadMacros( | ||
| 30 : | "PGstandard.pl", | ||
| 31 : | "MathObjects.pl", | ||
| 32 : | "PGchoicemacros.pl", | ||
| 33 : | # "parserNumberWithUnits.pl", | ||
| 34 : | # "parserFormulaWithUnits.pl", | ||
| 35 : | # "parserFormulaUpToConstant.pl", | ||
| 36 : | # "PGcourse.pl", | ||
| 37 : | ); | ||
| 38 : | |||
| 39 : | Context("Numeric"); | ||
| 40 : | $showPartialCorrectAnswers = 1; | ||
| 41 : | |||
| 42 : | $a = random(1,5,1); | ||
| 43 : | $ad = ($a == 1) ? "" : $a; | ||
| 44 : | |||
| 45 : | $which = list_random(0,1,2); | ||
| 46 : | if ( $which == 0 ) { | ||
| 47 : | $ser = "$ad x + " . (4*$a) . " x^2 + " . (9*$a) . " x^3 + " . | ||
| 48 : | (16*$a) . " x^4 + " . (25*$a) . " x^5 + \cdots"; | ||
| 49 : | $r = Compute("1"); | ||
| 50 : | } elsif ( $which == 1 ) { | ||
| 51 : | $a = 2 if ( $a == 1 ); | ||
| 52 : | $ser = "$ad x + " . ($a*$a) . " x^2 + " . ($a**3) . " x^3 + " . | ||
| 53 : | ($a**4) . " x^4 + " . ($a**5) . " x^5 + \cdots"; | ||
| 54 : | $r = Compute("1/$a"); | ||
| 55 : | } elsif ( $which == 2 ) { | ||
| 56 : | $a-=2 if ( $a > 3 ); | ||
| 57 : | $ad = ($a == 1) ? "" : $a; | ||
| 58 : | $ser = "$ad x + " . (4*$a*$a) . " x^2 + " . (9*($a**3)) . " x^3 + " . | ||
| 59 : | (16*($a**4)) . " x^4 + " . (25*($a**5)) . " x^5 + \cdots"; | ||
| 60 : | $r = Compute("1/$a"); | ||
| 61 : | } | ||
| 62 : | |||
| 63 : | TEXT(beginproblem()); | ||
| 64 : | Context()->texStrings; | ||
| 65 : | BEGIN_TEXT | ||
| 66 : | |||
| 67 : | Use the ratio test to find the radius of convergence of the power | ||
| 68 : | series | ||
| 69 : | \[ $ser \] | ||
| 70 : | |||
| 71 : | \( R = \) \{ ans_rule(15) \} | ||
| 72 : | $BR | ||
| 73 : | ${BITALIC}(If the radius is infinite, enter ${BBOLD}Inf$EBOLD for R.)$EITALIC | ||
| 74 : | |||
| 75 : | END_TEXT | ||
| 76 : | Context()->normalStrings; | ||
| 77 : | |||
| 78 : | ANS( $r->cmp() ); | ||
| 79 : | |||
| 80 : | if ( $which == 0 ) { | ||
| 81 : | $soln =<<EOS; | ||
| 82 : | To find \(R\), we consider the following limit, where | ||
| 83 : | the coefficient of the \(n\)th term is given by | ||
| 84 : | \(C_{n} = $ad n^2.\) | ||
| 85 : | \[ | ||
| 86 : | \lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_{n}|} | ||
| 87 : | =\lim_{n\rightarrow\infty}\left|\frac{$ad (n+1)^2 x^{n+1}}{$ad n^2 x^n}\right| | ||
| 88 : | =\lim_{n \rightarrow \infty}|x|\frac{n^2+2n+1}{n^2} | ||
| 89 : | = |x|\lim_{n \rightarrow\infty}\left(\frac{1+({2/n})+({1/n^2})}{1}\right) | ||
| 90 : | = |x|. | ||
| 91 : | \] | ||
| 92 : | Thus, the radius of convergence is \(R=1\). | ||
| 93 : | EOS | ||
| 94 : | } elsif ( $which == 1 ) { | ||
| 95 : | $soln =<<EOS; | ||
| 96 : | To find \(R\), we consider the following limit, where | ||
| 97 : | the coefficient of the \(n\)th term is given by | ||
| 98 : | \(C_{n} = $a^n.\) | ||
| 99 : | \[ | ||
| 100 : | \lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_{n}|} | ||
| 101 : | =\lim_{n\rightarrow\infty}\left|\frac{$a^{n+1} x^{n+1}}{$a^n x^n}\right| | ||
| 102 : | =\lim_{n \rightarrow \infty}|x| a | ||
| 103 : | = a |x|. | ||
| 104 : | \] | ||
| 105 : | This must be less than one, so \(|x| < 1/a\), and our radius of convergence | ||
| 106 : | is \(R=1/$a\). | ||
| 107 : | EOS | ||
| 108 : | } elsif ( $which == 2 ) { | ||
| 109 : | $r = ( $a == 1 ) ? 1 : "1/$a"; | ||
| 110 : | $soln =<<EOS; | ||
| 111 : | To find \(R\), we consider the following limit, where | ||
| 112 : | the coefficient of the \(n\)th term is given by | ||
| 113 : | \(C_{n} = $a^n n^2.\) | ||
| 114 : | \[ | ||
| 115 : | \lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_{n}|} | ||
| 116 : | =\lim_{n\rightarrow\infty}\left|\frac{$a^{n+1} (n+1)^2 x^{n+1}}{$a^n n^2 x^n}\right| | ||
| 117 : | =\lim_{n \rightarrow \infty}|x| a\frac{n^2 + 2n + 1}{n^2} | ||
| 118 : | =\lim_{n \rightarrow \infty} a |x| \frac{1 + 2/n + 1/n^2}{1} | ||
| 119 : | = a |x|. | ||
| 120 : | \] | ||
| 121 : | This must be less than one, so \(|x| < 1/a\), and our radius of convergence | ||
| 122 : | is \(R=$r\). | ||
| 123 : | EOS | ||
| 124 : | } | ||
| 125 : | |||
| 126 : | Context()->texStrings; | ||
| 127 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 128 : | $PAR SOLUTION $PAR | ||
| 129 : | |||
| 130 : | $soln | ||
| 131 : | |||
| 132 : | END_SOLUTION | ||
| 133 : | Context()->normalStrings; | ||
| 134 : | |||
| 135 : | ENDDOCUMENT(); |
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