Alternating Series Tests: Understanding Conditional & Absolute Convergence

[twisted079]

I have a question about the ratio test. Suppose it proves inconclusive, we must than use another test to check for conditional convergence - 1) this test has to be associated with an alternating series, such as the Alternating Series Test, correct? (we wouldn't be able to use something like Integral Test, right?)
2) I noticed in the examples we use Alternating Series Test, if this test proves conditionally convergent, I know it doesn't prove Absolute Convergence. So now we would have to check for Absolute Convergence, correct? *** In this case, we can use tests such as the Comparison Test, which neglects that it is an alternating series? So basically at this point we take the absolute value of the series, which neglects any negatives, leaving it open for other tests?

 

[susskind_leon]

If the ratio test proves inconclusive, it tells you nothing, so you have to find another way to prove convergence or divergence.

If you have an alternating series, things are usually pretty easy (because of the Alternating series test).

If you want to prove abolute convergence or divergence, the alternating series test doesn't help you. If you want to prove absolute convergence, you must prove the convergence of the absolute series. So you can try, as you already mentioned, the comparison test or the integral test or whatever other test you have at hands.

 

[twisted079]

Thank you for taking the time out to reply, your answer helped clear my misconceptions.

 

[susskind_leon]

No worries mate! Glad I could help. You might want to check wiki for more convergence tests:
http://en.wikipedia.org/wiki/Convergence_tests

Cheers