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I have five kind of test here
1. Divergent test
2. Ratio test
3. Integral test
4. Comparison test
5. Alternating Series test

And a few questions here.
1. Are test 1,2,3,4 only available for Positive Series? and alternate series test is only for alternating series?
2. To show $\sum_{n=1}^{\infty}(-1)^n$ diverge I can't use alternating series test right? it just tell me the series doesn't converge. So, i tried to use divergence test? but it seems like edivergence test is not applicable for alternating series.

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I assume that for (1) you mean the theorem that says that if the $n^\text{th}$ term does not approach 0 as $n \to \infty$ then the series diverges. This test does not require the terms to be positive, so you can apply it to show that the series $\sum_{n=1}^\infty (-1)^n$ diverges.

The ratio test does not require the terms to be positive. You end up taking the absolute value in this test, so signs do not matter.

The usual formulations of the integral test and comparison test only apply to series with positive terms.

The alternating series test is only for alternating series, as the name suggests. It has a couple of other requirements also.

The alternating series test never tells you that a series diverges. If the hypotheses are met, then the conclusion is that the series converges.

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  • $\begingroup$ wow, thank you so much for clearing the concept $\endgroup$ – Timothy Leung Jun 4 '13 at 3:12
  • $\begingroup$ @Timothy I'm glad to have helped $\endgroup$ – Trevor Wilson Jun 4 '13 at 3:15
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The Test for Divergence works for alternating series. Might I also recommend a short read of this?

Also, you said that:

I can't use alternating series test right? it just tell me the series doesn't converge.

This isn't entirely correct. Failure of the AST does not mean the series does not converge (that would imply divergent); failure of the AST means no conclusions can be draw about the convergence/divergence of the series.

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  • $\begingroup$ Isn't that same as the divergent test? So, I can apply divergent test this case? $\endgroup$ – Timothy Leung Jun 4 '13 at 3:11
  • $\begingroup$ @TimothyLeung, I made a mistake in my names; yes, they are the same and you may use it. $\endgroup$ – user80696 Jun 4 '13 at 3:21

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