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11.8_ Power Series-solutions

# 11.8_ Power Series-solutions - stiurca(mas7745 11.8 Power...

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stiurca (mas7745) – 11.8: Power Series – meth – (91825) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points ±ind the interval oF convergence oF the se- ries s n =1 ( n ) n x n 2 n . 1. ( 2 , 2] 2. [ 1 , 1) 3. ( 1 , 1) 4. converges only at x = 0 correct 5. ( 1 , 1] 6. [ 2 , 2) Explanation: When a n = ( n ) n x n 2 n , it’s more convenient to use the Root Test to determine the interval oF convergence. ±or then | a n | 1 /n = v v v v ( n ) n x n 2 n v v v v 1 /n = n | x | 2 . But lim n →∞ n | x | 2 = For all x n = 0. By the Root Test, thereFore, the given series converges only at x = 0 . 002 10.0 points Determine the radius oF convergence, R , oF the series s n ( 1) n 6 n ( n + 2) x n . 1. R = 2 2. R = 6 correct 3. R = 0 4. R = 1 6 5. R = 6. R = 1 2 Explanation: The given series has the Form s n a n x n with a n = ( 1) n 6 n ( n + 2) . Now For this series, (i) R = 0 if it converges only at x = 0, (ii) R = if it converges for all x , while iF R > 0, (iii) it converges when | x | < R , and (iv) diverges when | x | > R . But lim n v v v a n +1 a n v v v = lim n n + 2 6( n + 3) = 1 6 . By the Ratio Test, thereFore, the given series converges when | x | < 6 and diverges when | x | > 6. Consequently, R = 6 .

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stiurca (mas7745) – 11.8: Power Series – meth – (91825) 2 003 10.0 points Find the interval of convergence of the power series s n =0 5 n 8 n 5 + 7 x n . 1. b 5 8 , 5 8 B 2. b 5 8 , 5 8 p 3. b 1 5 , 1 5 p 4. b 1 5 , 1 5 B correct 5. [ 5 , 5) 6. [ 5 , 5] Explanation: We ±rst apply the Ratio Test to the in±nite series s n 5 n 8 n 5 + 7 | x | n . For this series a n +1 a n = 5 n +1 5 n P 8 n 5 + 7 8( n + 1) 5 + 7 p | x | . But 8 n 5 + 7 8( n + 1) 5 + 7 = 8 + 7 n 5 8 P n + 1 n p 5 + 7 n 5 , so lim n →∞ a n +1 a n = lim n 5 | x | P 8 + 7 n 5 p 8 P n + 1 n p 5 + 7 n 5 = 5 | x | .
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11.8_ Power Series-solutions - stiurca(mas7745 11.8 Power...

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