Power Series

A power series is the most important special case of all series: it has the form $\sum a_n x^n$, its domain of convergence is an interval centred on the origin, and its structure is extremely regular.

The core of this chapter is finding the radius of convergence (by the ratio test or the root test) and determining the interval of convergence — at the two endpoints $x=\pm R$ you still have to check separately, and that is the step most often overlooked. The Taylor expansion then shows that as long as a function is smooth near the point of expansion, it can be approximated to any accuracy by a polynomial.

That polynomials can approximate an arbitrary function is a result whose importance is often underestimated. This chapter gives the expansions of several common functions and explains why they are so useful in computation and numerical analysis.