chap11sec5 - n S b-=-∑ is the sum of an alternating...

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11.5 Alternating Series An alternating series is a series whose terms are alternately positive and negative. They will often have the term (-1) n  as part of  the definition of the series. The test for determining convergence or divergence as follows: The Alternating Series Test If the alternating series  ( 29 1 1 2 3 4 1 0 n n n b b b b b b - - = - + - + L satisfies     1 ( ) n n a b b + for all n  and      ( ) lim 0 n n b b →∞ =      then the series is convergent. Example: Determine convergence or divergence for each series.       2 1 2 1 1 1 ( 1) ln ( ) ( ) ( 1) ( ) ( 1) 1 3 n n n n n n n n a b c n n n - = = = - - - + + Look at Example 1 on page 728. This series is the  alternating harmonic series  and it  is convergent. Estimating Sums We have a theorem to use when estimating the sum of a convergent alternating series. Theorem:     If  1
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Unformatted text preview: n S b-=-∑ is the sum of an alternating series that satisfies ( 29 1 1 ( ) 0 and lim then n n n n n n n i b b ii b R s s b + + →∞ ≤ ≤ = =-≤ Example: How many of the terms of the series 1 4 1 ( 1) n n n + ∞ =-∑ do we need to add in order to find the sum so that the error <0.001? Example: Approximate ( 1) (2 )! n n n ∞ =-∑ to four decimal places. Example: Find an approximation of the series 1 1 ( 1) n n n + ∞ =-∑ using the partial sum S 100 . What is the maximum possible error using this approximation? Example: Find an integer n such that, using S n as an approximation of the series 2 ( 1) ln n n n ∞ =-∑ the maximum possible error is 0.0001....
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