sec11_5

sec11_5 - n =1 (-1) n +1 n 1 Section 11.5.Alternating...

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11.5 Alternating Series 1. Alternating Series Test An alternating series is a series whose terms are alternately positive and negative. 1 - 1 2 + 1 3 - 1 4 + 1 5 - ... = ± n =1 ( - 1) n - 1 n In the form of a n =( - 1) n - 1 b n , or a n =( - 1) n b n where b n = | a n | . Theorem 1.1 (The Alternating Series Test) . If the alternating series ± n =1 ( - 1) n - 1 b n = b 1 - b 2 + b 3 - b 4 + b 5 - b 6 + ..., b n > 0 satisFes ( i ) b n +1 b n for all n ( ii ) lim n →∞ b n =0 then the series is convergent. Example 1.1 (The Alternating Harmonic Series) . Determine whether or not the series converges or diverges. Explain.
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Unformatted text preview: n =1 (-1) n +1 n 1 Section 11.5.Alternating Series 2 Example 1.2. (problem 4) Test the series for convergence or divergence. 1 2-1 3 + 1 4-1 5 + 1 6-... Example 1.3. (problem 11) Test the series for convergence or divergence. n =1 (-1) n +1 n 2 n 3 + 4 Example 1.4. (problem 15) Test the series for convergence or divergence. n =1 cos n n 3 / 4...
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This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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sec11_5 - n =1 (-1) n +1 n 1 Section 11.5.Alternating...

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