In which we meet three really important tests for convergence.
- Our motto for this part of the course is “Tests for convergence are an aid to thinking, not a substitute for thinking”.
- We reminded ourselves of the definition of absolute convergence.
- Theorem 9 (Absolute convergence implies convergence) If the complex series is absolutely convergent then it is convergent. We proved this first for the case that is real for all , by splitting into non-negative and negative terms, and then used this to deduce the result even when is complex.
- Theorem 10 (Ratio test) Let be a complex series with for all . Suppose that there is some real number such that as . If , then the series converges absolutely (and so converges). If , then the series diverges. We saw that this was essentially comparison with a geometric series. (The root test, on Examples Sheet 1, has a similar flavour to the ratio test.)
- Theorem 11 (Cauchy condensation test) Let be a decreasing sequence of positive numbers. Then converges if and only if converges. We proved one direction by collecting terms in blocks corresponding to powers of 2, and we’ll do the other direction similarly next time.
Understanding today’s lecture
- Keep adding these new tests and new examples to your series grids. Are you starting to get a feel for which tests are useful for which types of series?
- We said that the ratio test is inconclusive if the ratio tends to 1 or if the ratio does not have a limit. Find examples to explore this.
- The Cauchy condensation test was for series where is a decreasing sequence of positive numbers. What happens if the sequence isn’t decreasing, or the terms aren’t all positive? You should get into the habit of investigating whether conditions like this are necessary.
- Can you prove the other direction of the Cauchy condensation test?
There are lots of books and websites that explain these standard tests of convergence. If you come across any good examples, please do share recommendations in the comments below.
Also, if you find alternative proofs of any of the results in lectures then please leave a comment to tell others about it: it’s always good to have more than one argument (even if I don’t have time to give more in lectures).
This isn’t directly related to the Analysis I course, but hopefully might be interesting: a blog post about the size of Gauss sums, which links closely with this business of sums with and without cancellation (and which has some pictures).
Preparation for Lecture 6
For which is the series convergent?
We have seen that every absolutely convergent series is convergent. Are there any series that are convergent but not absolutely convergent?