Selected posts from my WordPress blog

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Since I can now use MathJax here, but not on my WordPress blog, I am going to transfer some of my recent posts over from WordPress, using MathJax here. But first I have a lot of marking to do!

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A rule of thumb for when to use the Ratio Test

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The Ratio Test is a powerful test for both sequences and series of non-zero real numbers. It comes in various forms, but here is one commonly used version. Theorem (Ratio Test) Let (xn)n∈N be a sequence of non-zero real numbers. Suppose that |xn+1||xn|→L as n→∞, where L is either a non-negative real number or +∞. (a) If L>1, then |xn|→+∞ as n→+∞, the sequence (xn) is divergent, and the series ∞∑n=1xn diverges. (b) If L∈[0,1), then the series ∞∑n=1xn is absolutely convergent (and hence convergent), and the sequence (xn) converges to 0. (c) If L=1, then the Ratio Test is inconclusive , and you need to use a different test . (The Ratio Test tells you nothing in this case about the convergence or otherwise of either the sequence or the series.) Notes Here (b) does not tell you anything about the value of the sum of the series ...
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An introduction to the Hahn-Banach extension theorem: Part VI

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In this post we sketch a second proof of the Hahn-Banach Extension Theorem for continuous linear functionals on real normed spaces, this time using transfinite induction. We omit some of the details, which the reader can fill in. In particular, some of the details are similar to those involved in proving the claim made in Part V (when we obtained an upper bound for a chain). For a non-zero ordinal γ, recall that, with ordinal interval notation, we have γ=[0,γ). That is, γ is equal to the set of all ordinals which are strictly less than γ. However, we use ordinal interval notation throughout. If you prefer to use γ instead of [0,γ) that is fine, of course! Looking at the Wikipedia page on the Hahn-Banach Theorem , apparently both Hahn and Banach (independently) used transfinite induction in their proofs, rather than Zorn's Lemma. Theorem (Hahn-Banach Extension Theorem for linear functionals on normed spaces over R)  Let (E,‖⋅‖) ...
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An introduction to the Weierstrass M-test: Part I

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In order to make these posts accessible to first-year undergraduate students, I am going to be working with real-valued functions defined on intervals in R. I'm also going to avoid focusing on uniform convergence for sequences of functions, and instead focus on absolute convergence of series of functions, and how this fits with continuity. I'll then start to look at differentiability. For those interested in my approach to teaching uniform convergence for sequences of functions, I have a video about this on YouTube (at  https://www.youtube.com/watch?v=mB4Yny0T3HA ). Students watching this video should be warned that my non-standard notion of " absorption " (intended to help students understand convergence of sequences properly) has not really caught on! Other lecturers are unlikely to know about this notion of absorption, and so you should avoid using that terminology unless you explain it locally. However, if you want to know more about my absorption approach, s...
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