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Alternating Series Test

Alternating Series Test - Alternating Series Test The last...

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Alternating Series Test The last two tests that we looked at for series convergence have required that all the terms in the series be positive. Of course there are many series out there that have negative terms in them and so we now need to start looking at tests for these kinds of series. The test that we are going to look into in this section will be a test for alternating series. An alternating series is any series, , for which the series terms can be written in one of the following two forms. There are many other ways to deal with the alternating sign, but they can all be written as one of the two forms above. For instance, There are of course many others, but they all follow the same basic pattern of reducing to one of the first two forms given. If you should happen to run into a different form than the first two, don’t worry about converting it to one of those forms, just be aware that it can be and so the test from this section can be used. Alternating Series Test Suppose that we have a series and either or where for all n . Then if,
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1. and, 2. is a decreasing sequence the series is convergent. A proof of this test is at the end of the section. There are a couple of things to note about this test. First, unlike the Integral Test and the Comparison/Limit Comparison Test, this test will only tell us when a series converges and not if a series will diverge. Secondly, in the second condition all that we need to require is that the series terms, will be eventually decreasing. It is possible for the first few terms of a series to increase and still have the test be valid. All that is required is that eventually we will have for all n after some point. To see why this is consider the following series, Let’s suppose that for is not decreasing and that for is decreasing. The series can then be written as, The first series is a finite sum (no matter how large N
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