1. Power Series Chapter 12.8,12.9

2. Basic Definitions: A power series looks like: All the techniques for testing convergence apply but now we are interesting in the domain of x-values for which the series will converge  radius of convergence Power series allow us to extend our ideas of functions and infinite series Many important functions can be “re-written” as power series

3. Examples… Find the radius of convergence for Set up with ratio-test Evaluate limit Find domain of values for x which (if at all) ensure series passes the ratio test

4. Functions as Infinite Series… Carry out the following long division… You get which is a very versatile result! example: what is ?

5. Power Series and Differential Equations Historically, it was in the field of differential equations that power series emerged. Example: Solve the 2 nd order DE subject to the condition f(0) = 0. Since we don’t “know” anything about f(x) start with Fourier (1768-1830) was one of the earliest mathematical physicists to explore the rich connection between infinite series and differential equations