Testing convergence: alternating series test and ratio test

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Joined: Sun Oct 07, 2007 1:49 pm

Testing convergence: alternating series test and ratio test

Post by fullofquestions » Sun Oct 07, 2007 2:25 pm

I am using Stewart's Early Transcendental's 5E to brush up on Sequences and Series. Unfortunately, I am not seeing the difference between the alternating series test and the ratio test. I think the problem arises when I see that the sequence in the ratio test needs to approach a finite value (not necessarily 0) while the alternating series test does... I realize that the alternating series test is used to test alternating series (obviously) and entails:
1) shwowing that b(sub n + 1) =< b(sub n), i.e that the sequence is decreasing and
2) lim as n goes to infinity b(sub n) = 0

The ratio test is used for positively termed series, so I thought, but the notation in the book (p742) lists that:
lim as n goes to infinity ABS(a(sub n +1) / a(sub n)) = L
if L< 1, then the series is absolutely convergent
if L > 1, then the series is divergent
if L = 1 then the test is inconclusive

Since they are taking the ABSOLUTE value of the ratio, this implies that some of the terms are negative... Furthermore the example on page 743 is actually using the ratio test on an alternating series.

My question is that these tests look extremely similar and at the moment I'm not seeing how to determine which to use. Actually, by the way the book treats this matter I feel like I can just completely cross out one of these tests and keep just one of them to solve all questions of this type. In this regard this would be very helpful because ideally I would like to have a lean list of tests that I can use on the exam. I realize that with the right amount of tests I should be able to answer all questions without much loss for efficiency/time. So far, I found the following site that is has a very lean list of tests although I has not helped in answering my question. Hopefully this list will be useful to some

http://www.vias.org/calculus/09_infinit ... 06_10.html

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Joined: Mon Jul 23, 2007 12:41 am

Post by heybob » Thu Oct 11, 2007 11:24 am

Well, if you have an Alternating series, you can use the alternating series test to see if it converges. If it does, then try applying the Ratio Test i.e. take the absolute value of the series.

If it also converges, then the series is absolutely convergent, a stronger form of convergence.

For example the series [(-1)^n ] / ( n ) has decreasing terms and the limit goes to zero, right? So by the alternating test, it converges. However, consider the ratio test. Then you are checking if the series 1/n converges. You should know from the p-series, that 1/n always DIVERGES. Hence, the alternating series [(-1)^n ] / ( n ) converges but is NOT absolutely convergent.

You'll see if you apply the ratio test to 1/n, the limit approaches 1, meaning it's inconclusive. But you can use the integral test or something that proves it is indeed divergent.

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