alt_series

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 inFnite 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
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This note was uploaded on 04/02/2008 for the course MATH 31B taught by Professor Valdimarsson during the Fall '08 term at UCLA.

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

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