How Do Taylor Series Work?
Discussing what to know about the Taylor Series
The Taylor Series is perhaps one of the most interesting topics that is covered in a calculus class. Certainly for me, learning about the Taylor Series was a daunting task full of twists and turns. However, the process was extremely rewarding, and the knowledge gained proved to be game changing on my outlook of calculus in general. In this article, I hope to articulate some of the concepts and ideas that I gathered about the Taylor Series.
The idea surrounding the Taylor Series is mostly attributed to Brook Taylor, who conducted worked on the concept in the early 18th century. The series itself is an infinite series of derivatives at a point that models a function inside a given interval, the interval of convergence. As the highest power on a given term, or the order of the series, rises, the model better approximates the function as a whole. The Taylor Series does not approximate the function well outside of the interval of convergence.
Constructing the Series
A Taylor Series can be constructed at any x-coordinate, but there is a special name for a Taylor Series that is constructed at x = 0: the Mclaurin Series. We’ll start with this, and then generalize it to all Taylor Series. The formula of the Mclaurin Series of a function…
