Taylor Series

Abstract

After the Fundamental Theorem of the Calculus, Taylor’s theorem and the Taylor series form the most important theoretical and practical tool of the calculus. They certainly comprise the central concept of numerical analysis. In the last chapter we developed many familiar functions in series and acquired some facility in their use. We shall now study some applications of this theory. A first Example develops a series that approximates the logarithm function at x=2, even though the function is not defined at all at x=0. Next we describe Newton’s method and give Examples of its use and misuse for the functions ex-2x-1 and (x-1)/x2. Then series integration is explored with the Sine and Fresnel integrals as Examples. In the Example of 1/(1-x2) we discuss and then analyze the error in series integration.