Analysis I: Lecture 6

In which we think about series that are convergent but not absolutely convergent.

• We finished proving the Cauchy condensation test (Theorem 11).
• We saw the key example that $\sum\limits_{n=1}^{\infty} \frac{1}{n^{\alpha}}$ converges if and only if $\alpha > 1$.  We proved this using the Cauchy condensation test.
• Theorem 12 (Alternating series test) Let $(a_n)_{n=1}^{\infty}$ be a decreasing sequence of non-negative real numbers such that $a_n \to 0$ as $n \to \infty$.  Then $\sum\limits_{n=1}^{\infty} (-1)^{n+1} a_n$ converges.  We proved this by considering two subsequences of the partial sums, $(s_{2N})_{N=1}^{\infty}$ and $(s_{2N+1})_{N=1}^{\infty}$,  We showed that the first was monotone and bounded, so convergent.  Then we showed that the second tends to the same limit, and then that the whole sequence tends to that limit.
• We discussed the idea that the proofs of these tests are at least as important as the tests themselves: they are worked examples of strategies for showing that series converge or diverge.
• We saw the definition of conditional convergence, and mentioned that we have to be careful when dealing with conditionally convergent series: we can’t rearrange terms at will.  There is a question on Examples Sheet 1 that relates to this.
• Theorem 13 Let $\sum\limits_{n=1}^{\infty} a_n$ be an absolutely convergent complex series.  Let $\sum\limits_{n=1}^{\infty} a_n'$ be a rearrangement of the series.  Then $\sum\limits_{n=1}^{\infty} a_n'$ also converges, and $\sum\limits_{n=1}^{\infty} a_n' = \sum\limits_{n=1}^{\infty} a_n$.  That is, if a series is absolutely convergent then we can safely sum the terms in any order we like.  We proved this in the case that all the terms are real and non-negative.  To extend it to the case that all terms are real, we could split into non-negative and negative parts as we did in the proof that absolute convergence implies convergence (Theorem 9), and then we could extend it further to complex terms.  This is an exercise.
• This is the end of this section on sequences and series.  We’ll return to these ideas later in the course, but we’ll move on to continuity next time.  You should now have all the tools that you need for the first examples sheet.

Understanding today’s lecture

• Make sure you keep updating your series grid.  Do you have a tick and a cross in every row?
• For $n \geq 1$, let $a_n = \frac{1}{\sqrt{n}} + \frac{(-1)^{n-1}}{n}$.  Show that each $a_n$ is positive, and that $a_n \to 0$ as $n \to \infty$.  Does $\sum\limits_{n=1}^{\infty} (-1)^{n-1}a_n$ converge?  What does this tell us about the alternating series test?
• Do we need all of the conditions in the alternating series test?  If we drop any one of them, is the result still true?
• Which of the convergent series in your grid are absolutely convergent, and which are conditionally convergent?
• Can you finish proving Theorem 13?

There are many other tests for convergence that we haven’t discussed.  We’ll see at least one more later in the term.  I recommend thinking of the tests that we’ve seen in lectures more as worked examples than as theorems: sometimes the conditions for a test won’t be satisfied, but you’ll be able to adapt the idea of the proof to your specific example.  The Tricki has some general advice about what to do if you’re trying to show that a sequence converges, and what to do if you’re thinking about an infinite series.  Did you know all that advice already?

Preparation for Lecture 7

• What does it mean to say that a function is continuous?  Try some examples of functions that (intuitively) are continuous and functions that aren’t, to get a feel for what the concept means.  Can you formulate a definition phrased in terms of limits?  Can you formulate a definition using $\epsilon$ and $\delta$?
• Why does every cubic polynomial with real coefficients have (at least) one real root?  Can you generalise this idea?