Calculus pp 300-319

# Power Series

• Harley Flanders
Part of the Textbooks in Mathematical Sciences book series (TIMS)

## Abstract

Power series generalize polynomials. A polynomial F(x) is a finite sum
$$F\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + \cdots + {a_n}{x^n}$$
A power series is an infinite series
$$F\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + \cdots + {a_n}{x^n} + \cdots = \sum\limits_{n = 0}^\infty {{a_n}{x^n}}$$
which, under suitable convergence conditions, defines a function F(x). In this chapter we study such power series (centered at x = 0) and also power series of the form
$$F\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{(x-c)^n}}$$
(centered at x = c). For any particular value of x, the series is an infinite series of numbers, which we know all about. We shall soon see that the series converges on an interval centered at c. There the power series defines a function.

## Keywords

Power Series Taylor Series Taylor Expansion Infinite Series Successive Term
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Harley Flanders
• 1
1. 1.University of MichiganAnn ArborUSA