Given some alternating series, the first step is to check whether it's absolutely convergent. Say it's not. Then you use the alternating series test. That test tells you if the series is convergent, but not necessarily if it's divergent (I think). So if it doesn't meet the conditions for that, what is your recourse? How do you confirm whether the series is conditionally convergent or not? All of the other tests I know require all positive terms -- so I don't know any other test to use.
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