I thought the muddiest part of the class on Thursday was the second part of the proof for showing that . The reason why I thought this was the muddiest point is because it is easy to confuse the difference between the upper-bound and the least upper-bound of a set and that once you have […]
In class, we worked toward defining the set of real numbers by investigating suprema. Vaguely speaking, we want to fill the “gaps” in the rational numbers. Simply adding in solutions to equations like is not sufficient (this yields the algebraic numbers, which still have “gaps”). Instead, we rely on the assumptions that (1) the rationals […]
In this post, I will discuss the following definitions we covered in the class: Upper Bound, Least Upper Bound (Supremum), and Axiom of Completeness (AoC). I will try to provide a proof template for most of the definitions I state. 1. Upper Bound If set A is a subset of ℝ, then a number b is called an […]
Three individuals will be chosen at the beginning of each class to update these three daily blog categories. Students providing updates to the category of “muddiest point” should make an attempt to share with their peers content from the class meeting that was most confusing to either themselves or their peers. The objective here is […]
Three individuals will be chosen at the beginning of each class to update these three daily blog categories. Students updating the “Definitions” category, should take the opportunity to engage with important definitions from the weekly readings and lectures. To truly engage with a definition, you are expected to go beyond simply copying down what is […]
Three individuals will be chosen at the beginning of each class to update these three daily blog categories. Included in “What Happened” should be a brief synopsis of the classroom activities: Who presented what problems, Highlights from the lecture notes, Observations from homeworks and exams, etcetera. This should not be a complete copy of your […]
WordPress allows you to use a lot of latex typesetting capabilities in this blog. To enter into “math mode,” when typing a post or a comment, use the command ‘dollar sign’latex. After typing this, you can use a large portion of the latex library of mathematical typesetting commands. When you are done writing mathematics, close math mode […]
Blogging has gained in popularity drastically in recent years, and mathematics-oriented blogs are likewise growing in both content and variety. Academics of all specializations are finding an active audience for their current, past, and future research endeavors. A blog is also a great place to explore ideas that may pertain to research, teaching, and learning, without being […]
Taylor Series Taylor Series provide a way to express many functions that are infinitely differentiable on some interval as a power series. If is such a function, then the power series coefficients are given by Taylor’s Formula which says that if , then . This is discussed in Chapter 6 of Abbott’s text. Let’s use this idea […]
Sequences of Functions We have developed our knowledge of sequence of number in the chapter 2. Now we would like to talk about knowledge of sequences of functions and their application in the real life. First lets look at the definition of sequences of functions. Suppose and for all there is a function . The […]
You must be logged in to post a comment.