In which we consider links between series and integrals.
- Definition of what it means for the integral to converge and diverge.
- Lemma 63: Let be a function such that is Riemann integrable on for each , and such that for all . Then exists if and only if there is a constant such that for all . We saw that this was analogous to the result that an increasing sequence, bounded above, converges (Theorem 1), and indeed we used that fact to prove the interesting direction of this lemma.
- Theorem 64 (Integral test, or integral comparison test): Let be a decreasing function with for all . Then and either both converge or both diverge. We proved this by drawing a picture and carefully thinking about increasing bounded sequences and Lemma 63.
- We used the integral test to consider the examples of and . We’d previously thought about series like this using the Cauchy condensation test.
Understanding today’s lecture
You could pick some examples of functions and consider whether the integral converges.
You could add the integral test to your series grid. How many of your examples of series can you study successfully using the integral test? In the other direction, are there any interesting integrals that you can study as a result of the integral test?
I’ve gone through the feedback questionnaires, and here are the books that you collectively mentioned (in alphabetical order by surname of author).
Baylis What is mathematical analysis?
Burkill A first course in mathematical analysis
Hardy A course of pure mathematics
Preparation for Tripos exams
There are loads of past Tripos papers online. When you’ve tried some of them, you might like to read some of Tim Gowers’s blog posts in which he discusses how to go about tackling such questions; he takes his examples from 2003 (so you should try those questions before reading the posts!).
Other exciting things sometimes appear on the Maths at Murray Edwards Facebook page.