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| 1 : | aubreyja | 2584 | # DBsubject('Calculus') |
| 2 : | jj | 3224 | ## DBchapter('Infinite Sequences and Series') |
| 3 : | aubreyja | 2584 | # DBsection('Representations of Functions as Power Series') |
| 4 : | # KEYWORDS('calculus', 'series', 'sequences', 'power series', 'convergence', 'radius of convergence', 'interval of convergence') | ||
| 5 : | # TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | # EditionText1('2') | ||
| 7 : | # AuthorText1('Rogawski') | ||
| 8 : | # Section1('10.6') | ||
| 9 : | # Problem1('15') | ||
| 10 : | # Author('Emily Price') | ||
| 11 : | # Institution('W.H.Freeman') | ||
| 12 : | DOCUMENT(); | ||
| 13 : | |||
| 14 : | |||
| 15 : | |||
| 16 : | #Load Necessary Macros | ||
| 17 : | |||
| 18 : | loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", ); | ||
| 19 : | loadMacros("Parser.pl"); | ||
| 20 : | loadMacros("freemanMacros.pl"); | ||
| 21 : | |||
| 22 : | |||
| 23 : | Context()->variables->add(n=>'Real'); | ||
| 24 : | |||
| 25 : | #Book Values | ||
| 26 : | #$exponent = 2; | ||
| 27 : | |||
| 28 : | $exponent = random(2, 9); | ||
| 29 : | |||
| 30 : | Context()->texStrings; | ||
| 31 : | |||
| 32 : | BEGIN_TEXT | ||
| 33 : | \{ beginproblem() \} | ||
| 34 : | \{ textbook_ref_exact("Rogawski ET 2e", "10.6", "15") \} | ||
| 35 : | $PAR | ||
| 36 : | Find the interval of convergence for the following power series: | ||
| 37 : | $PAR | ||
| 38 : | \[ \sum_{n=1}^{\infty} \frac{x^n}{(n!)^{$exponent}} \] | ||
| 39 : | $PAR | ||
| 40 : | The interval of convergence is: \{ans_rule() \} | ||
| 41 : | END_TEXT | ||
| 42 : | |||
| 43 : | |||
| 44 : | Context()->normalStrings; | ||
| 45 : | |||
| 46 : | #Answer Check Time! | ||
| 47 : | ANS(Interval("(-inf, inf)")->cmp); | ||
| 48 : | |||
| 49 : | Context()->texStrings; | ||
| 50 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 51 : | $PAR | ||
| 52 : | $SOL | ||
| 53 : | With \( a_n = \frac{1}{(n!)^{$exponent}}\), | ||
| 54 : | \[ \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{((n+1)!)^{$exponent}} \cdot \frac{(n!)^{$exponent}}{1} = \left( \frac{1}{n+1} \right)^{$exponent} \] | ||
| 55 : | and | ||
| 56 : | \[ r = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0. \] | ||
| 57 : | The radius of convergence is therefore \(R = r^{-1} = \infty\), and the series converges absolutely for all \(x\). Thus, the interval of convergence is \((-\infty,\infty) \). | ||
| 58 : | |||
| 59 : | END_SOLUTION | ||
| 60 : | |||
| 61 : | ENDDOCUMENT() |
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