Lecture 8-6 - to approximate the sum. 3 b) Approximate the...

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8.6 - Alternating Series and Absolute and Con- ditional Convergence Defn. Alternating Series A series in which the terms alternate between positive and negative is called an alternating series . Classic Example: The Alternating Series Test (AST) X n =1 ( - 1) n a n converges if ALL of the following conditions hold. 1. 2. 3. 1
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Examples Determine if the following series converge or diverge. 1. X n =1 ( - 1) n +1 1 n 2. X n =1 ( - 1) n +1 n n + 1 Note: 2
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Theorem. Alternating Series Estimation Theorem If s = ( - 1) n +1 a n is the sum of an alternating series that satisfies the conditions of the AST, then the partial sum s n approximates the sum of the series with Example Consider the alternating series X n =1 ( - 1) n +1 n n 3 + 1 . a) Approximate the error involved in using the first 5 terms of the series
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Unformatted text preview: to approximate the sum. 3 b) Approximate the sum of the series so that the error is at most 0.01. 4 Absolute and Conditional Convergence Defn. Absolutely Convergent A series a n is called absolutely convergent if Defn. Conditionally Convergent A series a n is called conditionally convergent if Theorem. If a n converges absolutely then a n converges. Tip: 5 Examples Determine if the following series converge absolutely, converge conditionally, or diverge. 1. X n =1 (-1) n +1 n 2 2. X n =1 (-1) n +1 3 n 6 3. X n =1 (-1) n ( n !) 2 3 n (2 n + 1)! 4. X n =1 (-1) n n 2 2 n + 1 7...
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Lecture 8-6 - to approximate the sum. 3 b) Approximate the...

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