1. 11.8 Power Series

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

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

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

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

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

7. General case For a given power series there are three possibilities:
1.) The series converges for all values of x
2.) The series only converges for x=a
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
(R=infinity)
(R=0)

8. 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!

9. 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

10. Closer look at YES! Can we use this to express other functions as
power series?

11. Example
Express
as a power series.

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