Notes on the Alternating Series Test

Notes on the Alternating Series Test - Notes on the...

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Notes on the Alternating Series Test The Alternating Series Test applies only to a series with a very special form. Namely, the series must be of the form n a , where 1 0 n n a a + < . That is, the terms of the series alternate between positive and negative values. The test is relatively easy to apply and one can estimate the error between the actual sum of the series and a partial sum of the series. The Alternating Series Test : Let ( 29 1 n n b - be an alternating series with 0 n b . The series converges if a) 1 n n b b + for all n , and b) lim 0 n n b →∞ = . There are several observations one must keep in mind in applying the Alternating Series Test. A. This is the first test we have that applies to a series where the terms are not eventually all positive. But one must identify the terms n b so that they are positive. B. One must verify Condition (a) when applying this test. While the test states the inequality holds for all n , it need only hold for all n sufficiently large
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