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sn9_1 - Math 1432 Notes session 9 DAILY EMCF 9 1 Determine...

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Math 1432 Notes – session 9 DAILY EMCF 9 1. Determine whether the series converges or diverges. a) diverges b) converges c) cannot be determined 11.4 Absolute Convergence and Alternating Series An alternating series is a series whose terms alternate in sign. For example: ( 29 1 1 1 1 1 1 1 1 1 2 4 8 2 n n n - - = - + - + = - Alternating Series Test: If an alternating series ( 29 ( 29 1 1 2 3 4 1 1 , 0 n n n n a a a a a a - = - = - + - + satisfies ( i ) 1 n n a a for all n (non increasing) AND ( ii ) lim 0 n n a →∞ = then the series is convergent. Examples: 1. ( - - 1 1 2 n n
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2. ( 29 1 3 1 1 1 3 n n n n + = - + + 3. ( 1 n - If an alternating series converges, it can be classified as either absolutely convergent or conditionally convergent. If the series n a is convergent, then n a is convergent. We say that n a is absolutely convergent . If the series n a is convergent and the series n a is divergent, we say that n a is conditionally convergent . Examples: 1. ( - - 1 1 2 n n
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2. 2 sin( ) k k k π 3. ( 29 2 1 1 n n n - - 4. ( ( 1 1 k k k - + -
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If a convergent alternating series satisfies the condition 0 < a n+1 < a n , then the remainder R N involved in approximating the sum S by S N is less in magnitude than the first neglected (truncated) term. That is, 1 N N N R S S a = - Examples: 1. Approximate the sum of ( 29 1
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