Is there any particular method to use when dealing with these? Let me explain my problem. For example:
Here are the rules we know:
1. The Alternating Series Test
If the series
converges if all three of the following conditions are satisfied:
a. is positive for all n.
b. for all , for some integer N.
2. Absolute Convergence
Given if converges, we say that converges absolutely.
3. Conditional Convergence
If diverges, but converges, then we say that converges conditionally.
So where do I start with an alternating series? In this instance, we know is an increasing function, so it fails part b of the alternating series test. Also approaches one as n goes to infinity. So this alternating series fails the alternating series test completely. So therefore it diverges?
What about this one? I think this one meets of the conditions of the alternating series test. So then, if a series meets all the conditions, do I then use the conditional and absolute convergence tests? or do I just leave it at "converges by alternating series test" What is a good rule of thumb for these series?