104hw8sum06 - MATH 104, SUMMER 2006, HOMEWORK 8 SOLUTION...

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Unformatted text preview: MATH 104, SUMMER 2006, HOMEWORK 8 SOLUTION BENJAMIN JOHNSON Due July 31 Assignment: Section 23: 23.1(d), 23.7, 23.8 Section 24: 24.1, 24.9, 24.17 Section 23 23.1 Find the radius of convergence and determine the exact interval of convergence. (d) n 3 3 n x n Let a n = n 3 3 n . Then a n + 1 / a n = ( n + 1) 3 3 n + 1 / n 3 3 n = ( n + 1) 3 3 n 3 n + 1 n 3 = 1 3 n + 1 n ! 3 From limit laws, we have lim n | a n + 1 a n | = 1 3 . So the radius of convergence, R , is 3. to determine whether the interval of convergence is open closed, or half-open, we need to consider what happens to the series at x =- 3 and x = 3. For x =- 3, we have n 3 3 n (- 3) n = (- 1) n n 3 which diverges. Similarly, n 3 3 n (3) n = n 3 which also diverges. So the interval of convergence is (- 3 , 3). 23.7 For each n N , let f n ( x ) = (cos x ) n . Each f n is a continuous function. Nevertheless, show that (a) lim f n ( x ) = 0 unless x is a multiple of , Proof. If x is not a multiple of , then | cosx | < 1. So by one of the limit theorems, lim1....
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104hw8sum06 - MATH 104, SUMMER 2006, HOMEWORK 8 SOLUTION...

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