In the sections covered in this unit, we do two things. First, we examine more convergence tests on positive (or non-negative) series. The tests range from ugly to beautiful, but are all useful. In the second section, we open up to the idea of allowing negative terms in our series and find that this leads to some very odd complications.
- Section 9.5: In this section, we learn four closely-related convergence tests. The Comparison Test sets the stage for the proofs of the other three and is occasionally necessary (but often rather ugly). The Limit Comparison and Ratio Tests are rather neat-and-tidy - they tend to be the tests that you will go to first, they are easy to use, and still have a tremendous amount of computational power.
- Section 9.6: In this section, we find that when we allow series to take on both positive and negative values, things get very interesting. We explore some of the more far-reaching consequences in a video about math magic, but the amount needed for class is rather lower.