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9.6_LA_solutions - MthSc 108 9.6 Solutions(Alternating...

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MthSc 108 – 9.6 Solutions (Alternating Series, Absolute and Conditional Convergence) 1. Use the alternating series test to show that the series ( - 1) n + 1 ln n n n = 1 converges. The alternating series test applies because we have an alternating series - 1 ( ) n - 1 b n n = 1 = b 1 - b 2 + b 3 - b 4 + b 5 - b 6 + L where b n > 0 . Identify b n : - 1 ( ) n - 1 ln n n = ln n n Let f ( x ) = ln x x . a. Determine if lim n →∞ b n = 0 : Using L’Hospitals lim x →∞ ln x x = lim x →∞ 1 x = 0. b. Find the integer N , if there is one, for which b n + 1 b n : If f ( x ) = ln x x then f '( x ) = 1 - ln x x 2 < 0 when x > e , which means that f(x) is decreasing for x > e , which implies that b n + 1 b n for N 3 . Based on these two conditions, the series converges by the Alternating Series Test. 2. Answer the following questions using the convergent alternating series ( - 1) n n 4 n n = 1 . a. Find the 4 th partial sum of the series as an exact value. s 4 = ( - 1) 1 1 4 1 + ( - 1) 2 2 4 2 + ( - 1) 3 3 4 3 + ( - 1) 4 4 4 4 = - 5 32 b. How accurate is the approximation from part (a) for the sum of the series?
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