How Do Taylor Series Work?
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 f with derivatives of all orders is
As you may notice, the series consists of terms that are increasing in the degree of their derivatives. For example, the second term contains the first derivative of the function, the third term contains the second derivative of the function, the fourth term contains the third derivative of the function etc. Of course, the first term of the series contains no derivative.
Imagine for a moment that we cut off everything but the first two terms. Does this look familiar? It is actually in fact the equation for the tangent line of a function f at x = 0. The f(0) part takes care of the initial point, while the f’(0) part takes care of the instantaneous rate of change. Add on the third term, and we take care of the instantaneous concavity, or the rate of change of the rate of change. With each additional term, another “degree of the rate of change” is covered by the series, better modeling the function in…