alt_series - (12.5 Alternating Series Here we consider...

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(12.5) Alternating Series Here, we consider series which follow a particular pattern: the terms “alternate” in sign (positive, negative). It turns out it is often easy to verify if these converge: Alternating Series Test. Suppose a n > 0 for all n , the sequence { a n } is decreasing (that is, a n +1 a n for all n ), and lim n →∞ a n = 0. Then the infinite series n =1 ( - 1) n +1 a n converges. Example. n =1 ( - 1) n +1 1 n = 1 - 1 2 + 1 3 - 1 4 + 1 5 -· · · . Here, a n = 1 n is a positive, decreasing sequence whose limit is 0. So by the alternating series test, the series converges. (To the number ln 2 = 0 . 693 · · · , by the way.) The basic idea is that while the harmonic series n =1 1 n diverges (the partial sums gradually get larger and larger), the terms in the alternating harmonic series tend to cancel each other, so that can’t happen. Example. n =1 ( - 1) n +1 2 n 3 n - 1 . Here, a n = 2 n 3 n - 1 is positive and decreasing, but lim n →∞ 2 n 3 n - 1 = 2 3 . So we cannot use the alternating series test to show this series converges.

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• Spring '08
• Mathematical Series, series test

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