Explaining maths

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 ...

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...