11 8 Power Series Power Series We now

11. 8 Power Series

Power Series We now consider infinite series of the form: a is a constant c’s are coefficients that depend on n x is a variable

Example Note here that a = 0 and cn = 1 (for each n) Need |x|<1 For what values of x does the series converge?

Example Note here that a = 0 and cn = 1 (for each n) Sum = 1/(1 -x) If the series converges, what is the sum?

Example We just showed that WOW! This function can be represented as an infinite series If |x|<1

Example For what values of x do the following series converge?

General case For a given power series there are three possibilities: 1. ) The series converges for all values of x (R=infinity) 2. ) The series only converges for x=a (R=0) 3. ) There is a positive number R, called the radius of convergence, such that the series converges if |x-a|<R and diverges if |x-a|>R

Radius of Convergence • Use ratio or root test to find R • At the endpoints x=a+R and x=a-R, anything can happen! The series may converge or diverge…further testing must be done!

Why do we care? ? ? • We can represent functions as infinite power series (11. 9 functions as power series) • Note that a power series is an infinite polynomial defined by the coefficients cn • We like polynomials – Easy to integrate and differentiate

Closer look at Can we use this to express other functions as power series?

Example Express as a power series.

Coming Soon! • What about other functions such as Can we express these as power series? ? ?