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_infinite_series_06_10.html