Posts

An introduction to the Weierstrass M-test: Part IV

I think this will be the last post in this series. This time we will look at the special case of power series, using Theorem 5 from the previous part to help us to justify differentiation term by term. Notation:  Recall that we denote the set  \(\N \cup \{0\}\) of  non-negative  integers by \(\N_0.\) Recall the statement of our Theorem 5. Theorem 5 (Term by term differentiation of series) Let $I$ be a nondegenerate interval in $\R$, and let $M_n$ ($n \in \N_0$) be non-negative real numbers such that $\sum_{n=0}^\infty M_n < \infty\,.$ Let $f_n$ ($n \in \N_0$) be differentiable functions from $I$ to $\R,$ and suppose that, for all $n \in \N_0$ and all $x \in I,$ we have $|f_n'(x)|\leq M_n.$ Suppose further that, for  all $x \in I$, the series $\sum_{n=0}^\infty f_n(x)$ converges. Define $f:I \to \R$ by \[ f(x)=\sum_{n=0}^\infty f_n(x)\,.\] Then $f$ is differentiable on $I$, and, for each $x \in I$, \[f'(x)=\sum_{n=0}^\infty f_n'(x)\,.\] We are now ready to prove our main

An introduction to the Weierstrass M-test: Part III

In this post, following on from the first two posts in the series, we look at differentiating suitable series of functions term by term. We make as much use as possible of the Weierstrass M-test (in the spirit of this series of posts), instead of discussing uniform convergence in general. Of course, uniform convergence is an important topic, and I'll come back to it at some point. First we recall a characterization of differentiability in terms of "chord functions". Note, for functions defined on intervals, when we discuss differentiability at endpoints (if any) of the interval, we use one-sided derivatives. Proposition 3 (Differentiability in terms of chord functions) Let $I$ be a non-degenerate interval in $\R$, let $f:I\to \R$, and let $x_0 \in I.$ Then the following statements are equivalent. (a) The function $f$ is differentiable at $x_0$. (b)  There is a function $g:I \to \R$ such that $g$ is continuous at $x_0$ and, for all $x \in I$, \[f(x) = f(x_0) + (x-x_0) g(x

A Cauchy-L'Hopital-Taylor hybrid theorem

Last week I was teaching the standard theorems of real analysis (single variable): Rolle's Theorem, the Mean Value Theorem (MVT), Cauchy's Mean Value Theorem, L'Hôpital's rule, and Taylor's Theorem. As I was short of time, I didn't manage to give full proofs of everything live (but they have complete proofs in the printed notes). I did however have a think about the relationships between these results, and what I might say about them. Along the way, I noticed that there is a particular hybrid Cauchy-L'Hôpital-Taylor result which I don't remember seeing before (though no doubt it is well known). Let me start by recalling some of the standard statements. For the rest of this post , \(a\) and \(b\) are real numbers with \(a<b,\) and \(f\) and \(g\) are real-valued functions on \([a,b]\) which are both continuous on \([a,b]\) and differentiable on \((a,b).\) Rolle's Theorem Suppose that \(f(a)=f(b)\). Then there exists \(\xi \in (a,b)\) suc

An introduction to the Weierstrass M-test: Part II

In this part, we continue from where we left off in Part I of this series on the Weierstrass M-test . First, let's recall our standing assumptions and notation. Throughout, \(I\) is a non-degenerate interval in \(\R\), and \((f_n)_{n=0}^\infty\) is a sequence of functions from \(I\) to \(\R\).  We denote the set  \(\N \cup \{0\}\) of  non-negative  integers by \(\N_0.\) For each \(m \in \N_0\) we define the function \(S_m: I\to \R\) by \(S_m = \sum_{n=0}^m f_n\,.\) So, for each \(x \in I\), we have \(\Jdisplay S_m(x) = \sum_{n=0}^m f_n (x)\,.\) We also have a sequence \((M_n)_{n=0}^\infty\) of non-negative real numbers such that  \(\Jdisplay\sum_{n=0}^\infty M_n < \infty\,.\) For each \(m \in \N_0\), we set \(R_m=\sum_{n=m+1}^\infty M_n\,.\) Recall that \(R_m \to 0\) as \(m \to \infty,\) because the "tails of a convergent series" always tend to \(0.\) Our first task is to prove the following theorem that we stated last time. Theorem 1 (Part of the Weierstrass M-test)

An introduction to the Weierstrass M-test: Part I

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

Other approaches to connected components (from this year's MATH4085 Metric and Topological Spaces)

Hi everyone, In a previous post we looked at a sufficient condition for a union of connected sets to be connected. A relatively easy special case of Lemma 6.12 is the following. Lemma Let \(X\) be a (non-empty) topological space, and let \(E_1\) and \(E_2\) be connected subsets of \(X\) such that \(E_1 \cap E_2 \neq \emptyset\,.\) Then \(E_1 \cup E_2\) is also connected. Armed just with this lemma, we can define an equivalence relation   \(\sim\) on a non-empty topological space \((X,\tau)\) as follows: for \(x,y \in X\), \(x\sim y\) if and only if there exists a connected subset \(E\) of \(X\) such that both \(x\) and \(y\) are elements of \(E\). Let's check that this really is an equivalence relation. Since single point sets \(\{x\}\) are connected, reflexivity is easy. Symmetry is also immediate (it doesn't matter in which order you say that both \(x\) and \(y\) are elements of \(E\)). That leaves transitivity, which is where we use the lemma above. Let \(x,y,z \in X\) and

Conclusions on Totally Bounded metric spaces and Cauchy sequences

See also  https://math.stackexchange.com/questions/9964/if-no-cauchy-subsequence-exists-must-a-uniformly-separated-subsequence-exist  and  https://math.stackexchange.com/questions/7210/which-metric-spaces-are-totally-bounded  Let's list the conclusions from my recent posts. We'll use the same definitions and notation as before. These results are presumably all well known, though I haven't personally seen explicit statements of all of them before. I had some fun working out the details for myself! Notes on terminology : I have seen both "uniformly separated" and "separated" used for sequences. But there is a completely different notion of separated sets for pairs of sets. So possibly my use of the term "separated set" is confusing. I have seen the term "uniformly discrete" used, but I don't think that I have yet found the most definitive standard terminology. Theorem 1 Let \((X,d)\) be a metric space. Then  the following statements