The Taylor Series
Introduction and Summary
In this SparkNote, we introduce the theory of Taylor's formula, which arises in the attempt to approximate functions by polynomials. This approximation is a desirable thing to do, since polynomials are a good deal simpler than most functions we encounter. Polynomials are easier to evaluate at particular values and easier to differentiate and integrate, to name just a couple of advantages. The polynomials used to approximate a function are called Taylor polynomials; the power series with these polynomials as partial sums is called the Taylor series of the function.
We will first give an intuitive definition of the Taylor polynomials (and hence the Taylor series) in terms of derivatives. In order to get a handle on how well the Taylor polynomials approximate the actual function, we will have to come at them from another angle, making use of the integration by parts method. This will lead us to a careful study of the error (the remainder term) in the polynomial approximations. Finally, with the general theory under our belts, we will discuss a few of the most common Taylor series, and state a theorem on the differentiation and integration of Taylor series.