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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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So the direct cases are that: if it converges absolutely, it converges conditionally, or, simply if it passes alternating series test. (Alternating series test is often easy to use, so checking absolute convergence is not necessarily the first step. For instance, $\sum\frac{(-1)^n}{n}$) If it diverges for simple reason (say the terms don't go to $0$), then it does not converge absolutely.

Other cases may require some skills, and I don't think there is a general pattern. Abel's test is maybe one of the tests? Sometimes you can do rearrangement. You may check out this question for examples of alternating series failing alternating series test.

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