Analysis I: Lecture 4

In which we encounter our first couple of tests for convergence of series, and meet the series grid.

  • We proved Lemma 6 (ii), being careful to use partial sums.
  • We saw two key examples: the geometric series \sum\limits_{n=1}^{\infty} x^n, and the series \sum\limits_{n=2}^{\infty} \frac{1}{n(n-1)}.  We saw that in these two (rare cases) we could establish whether or not they converged by computing their partial sums.
  • Lemma 7 If the complex series \sum\limits_{n=1}^{\infty} a_n converges, then a_n \to 0 as n \to \infty.  This was easy, but will be useful.
  • Theorem 8 (Comparison test) Let a_1, a_2, a_3, .. and b_1, b_2, b_3, … be real numbers such that 0 \leq b_n \leq a_n for all n.  If \sum\limits_{n=1}^{\infty} a_n converges, then so does \sum\limits_{n=1}^{\infty} b_n.  We proved this by showing that the partial sums t_N = \sum\limits_{j=1}^N b_j form an increasing sequence that is bounded above.
  • We used the comparison test to show that \sum\limits_{n=1}^{\infty} \frac{1}{n^2} converges (comparing with \sum\limits_{n=2}^{\infty} \frac{1}{n(n-1)} from earlier).
  • We met the series grid.  I encourage you to create your own.  The rows correspond to tests of convergence, and the columns to examples of series. So as we meet a new series, or a new test of convergence, you should try the tests so far, or the series so far, to see which tests determine the convergence/divergence of which series, and put a tick or cross as appropriate.  (A tick means that the test can be used to show whether or not that series converges.)  You should ensure that each row contains at least one tick and one cross: so for each test you should have a series whose convergence or otherwise can be determined by that test, and another whose convergence or otherwise cannot be determined (if not, then you need to find a suitable example).
  • Definition of what it means for a series to be absolutely convergent.

Understanding today’s lecture

  • Make sure that you know why the converse of Lemma 7 is not true.
  • In Theorem 8, why did we need the terms to be non-negative?  It’s a good idea to get into the habit of asking yourself whether the conditions in a result are really necessary.
  • The series grid should be a good way of giving you practice at using the various tests we meet, and should also help you to become familiar with the properties of key examples.  Mathematicians rely on having collections of examples, so make sure that you start identifying your favourite illustrative series now!
  • You could start exploring examples of series to see whether or not they are absolutely convergent.  Perhaps you could look for some conventional examples (the ones that the person sitting next to you might think of) and some more exotic examples (that will be different from those found by the person sitting next to you).  This is a good way to get a feel for the extent of a definition: the things that are included and the things that aren’t.

Further reading

There’s an interesting Tricki article about how one might think of showing that \sum\limits_n \frac{1}{n^2} converges by comparing with \sum\limits_n \frac{1}{n(n-1)}.

I mentioned the book I want to be a Mathematician: an automathography by Paul Halmos — I recommend it.  Perhaps you can borrow a copy from the library?

Not directly related to this lecture, but nonetheless interesting: Tim Gowers has written about ‘just do it’ proofs.

And, completely irrelevantly, if you Google “”comparison test” series” and ask for pictures, then you get an interesting mix of mathematics and cars.  (For the record, I prefer the mathematics.)

Preparation for Lecture 5

  • We’re going to think about the relationship between convergence and absolute convergence, so over the weekend you could think about the last two questions from the blog post from Lecture 3 if you haven’t already done so.
  • Does the series \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n^2} converge?  What about the series \sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n}?
  • What can you say about whether or not the series \sum\limits_{n=1}^{\infty} \frac{n^{2013}}{2^n} converges?  Can you generalise your ideas?
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2 Responses to “Analysis I: Lecture 4”

  1. Analysis I: Lecture 5 « Theorem of the week Says:

    […] Expositions of interesting mathematical results « Analysis I: Lecture 4 […]

  2. Analysis I: Lecture 24 | Theorem of the week Says:

    […] 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 […]

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