1120_12 - Week 12 Outline Alternating Series Test...

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Unformatted text preview: Week 12 Outline Alternating Series Test Alternating Series Estimation Theorem Absolute Convergence and Conditional Convergence Power Series –Radius of Convergence and Interval of Convergence –Term-by-Term Differentiation and Integration Theorem –How to Find the Power Series Representation for a Function Definition An alternating series is a series whose terms are alternatively positive and negative. The Alternating Series Test If the alternating series ∑ ∞ n =1 (- 1) n +1 b n ( b n > 0) satisfies (1) b n +1 ≤ b n , for all n (2) lim n →∞ b n = 0, then the series is convergent. Remarks (a) If condition (2) does not hold, the series is of course divergent. (b) If condition (1) does not hold, the result may not be true. For example, ∑ (2- n- 3- n ) is convergent while ∑ (- 1) n ‡ 1 √ n + (- 1) n n · is divergent. Examples Determine whether the following series converge or diverge. 1. ∑ (- 1) n +1 n 2. ∑ (- 1) n n n +1 3. ∑ (- 1) n +1 n 2 n 3 +1 4. ∑( 1 3 n- 1 2 n ) 5. ∑ (- 1) n +1 √ n n +1 A partial sum s n of any convergent sequence can be used as an approximation to the total sum s . But this may not be quite useful unless we can estimate the accuracy of the 1 approximation. Alternating Series Estimation Theorem If s = ∑ ∞ n =1 (- 1) n +1 b n is the sum of an alternating series that satisfies (1) b n +1 ≤ b n , for all n (2) lim n →∞ b n = 0, then for n th partial sum s n we have | s- s n | ≤ b n +1 ....
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This note was uploaded on 12/09/2010 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell.

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1120_12 - Week 12 Outline Alternating Series Test...

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