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# notes9 - Tests for Convergence IV Last time we learned...

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Tests for Convergence IV Last time we learned about the Integral, p , comparison and limit comparison tests. The down side of these tests is that they can only be used if the terms a k in the series are not negative. The fist test we might be able to use when the a k ’s are both positive and negative is a test for when the terms of the series are alternating between + and - . For example, the series 1 - 1 / 2 + 1 / 3 - 1 / 4 + · · · = X n =1 ( - 1) n +1 n is an alternating series. Here a k = ( - 1) k +1 1 k which we could write as a k = ( - 1) k +1 b k where b k = 1 k to get a little more specific. Math 9C Summer 2011 (UCR) Pro-Notes October 12, 2011 1 / 9

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Theorem (The alternating series test) If the alternating series X k =0 ( - 1) k b k = b 0 - b 1 + b 2 - · · · satisfies I The b k ’s are decreasing, i.e. b k b k +1 for every k I lim k →∞ b k = 0 then X k =0 ( - 1) k b k is convergent. Math 9C Summer 2011 (UCR) Pro-Notes October 12, 2011 2 / 9
The alternating series test is a really easy test to use. If you’re given a series of the form X k =0 ( - 1) k b k all you have to check is that the b k ’s are decreasing and lim k →∞ b k = 0 and then you can say the series converges.

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notes9 - Tests for Convergence IV Last time we learned...

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