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 […]

Integration is generally thought of as finding the area between two curves, with usually one of the curves being the x-axis itself. The idea of integration, finding the area under the curve, is quite old. It can date back all the way to antiquity to Archimedes himself. Archimedes method is known as his method of […]

There are a great many topological spaces in the realm of mathematics, many of them quite exotic. Others are very similar to the systems we are familiar with. One of these is , the affinely extended real numbers The Definition of the Space The idea of the extended real numbers is to add a pair […]

Definition and Motivation In Chapter 6 of Understanding Analysis by Abbott, we explored some important properties of power series, where we tried to represent a function as a limit of polynomials. In this blog, we want to look at another type of powerful and widely-used series called Fourier series or trigonometric series, by representing a […]

The Challenge: Write a monosyllabic version of a proof we have studied this semester. Accepting the Challenge: My strategy was to choose a proof that was visually easy to understand and hope that I can convey the basic idea of the proof to the reader as an image. The proof I chose was the Nested Interval Property. […]

“What is the Fundamental Theorem of Calculus?…… Differentiation is easy; Integration is hard” – Ted Bunn In case you are wondering, no matter how true it might seem, that’s not the actual Fundamental Theorem of Calculus. Fortunately though, this post is primarily concerned with the birth of differential calculus. As I glance down at my prompt, it reads, […]

Over the past few weeks, we have talked about the continuity and differentiability of a function and we want to intuitively related these two concept with each other because they all characterize some important properties of a function. In Chapter 5 of Understanding Analysis by Abbott, there is a theorem states that if is differentiable […]

Formulations of Intermediate Value Theorem What is Intermediate Value Theorem? Let’s start with an interesting story I found online that can illustrate the result for Intermediate Value Theorem. One day a monk leaves at sunrise to climb up a mountain. He walks at a leisurely pace, sometimes stopping to enjoy the view, even retracing his […]

In Chapter 4, we spent much of our time trying to understand and prove continuity of functions. However, as we saw in Section 4.1, discontinuous functions can be even more interesting. Recall the Dirchlet Function, which we saw was discontinuous on all of , or well as the modified Dirchlet, which was discontinuous on , […]