Analysis I: Lecture 24

In which we consider links between series and integrals.

  • Definition of what it means for the integral \int_a^{\infty} f to converge and diverge.
  • Lemma 63: Let f : [a, \infty) \to \mathbb{R} be a function such that f|_{[a,R]} is Riemann integrable on [a,R] for each R > a, and such that f(x) \geq 0 for all x \in [a, \infty).  Then \int_a^{\infty} f(x) \mathrm{d}x exists if and only if there is a constant K such that \int_a^R f(x) \mathrm{d}x \leq K for all R > a.  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 f : [1, \infty) \to \mathbb{R} be a decreasing function with f(x) > 0 for all x \in [1, \infty).  Then \sum\limits_{n=1}^{\infty} f(n) and \int_1^{\infty} f(x) \mathrm{d}x 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 \sum\limits_{n=1}^{\infty} \frac{1}{n^{\alpha}} and \sum\limits_{n=3}^{\infty} \frac{1}{n (\log n)^{\alpha}}.  We’d previously thought about series like this using the Cauchy condensation test.

Understanding today’s lecture

You could pick some examples of functions f and consider whether the integral \int_a^{\infty} f 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?

Further reading

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

Spivak Calculus

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!).

Bonus things

Everybody knows that the end of term should end with videos.  So here are a couple of my favourites by Vi Hart: Infinity Elephants, and Stars.

Other exciting things sometimes appear on the Maths at Murray Edwards Facebook page.


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