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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 1709 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    1 # DBsubject('Calculus')
2 # DBchapter('Infinite Series and Sequences')
3 # 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
17
21
22
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()