1. Lecture 29 – Power Series Def: The power series centered at x = a:
x is the variable and the c’s are constants (coefficients)

2. For any power series, exactly one of the following is true:

3. Example 1 – Radius and Interval of Convergence
Ratio Test:
Series converges for

4. Example 2 – Radius and Interval of Convergence
Ratio Test:
Series converges for

5. Example 3 – Radius and Interval of Convergence
Ratio Test:

6. Example 3 – continued (testing endpoints)

7. Example 4 – Radius and Interval of Convergence
Root Test:

8. Example 4 – continued (testing endpoints)

9. Example 5 – Radius and Interval of Convergence
Geometric Series:

10. Example 5 – continued – what is the converging value?
Geometric Series:

11. Lecture 30 – More Power Series
The geometric series:
As a power series with a = 1, r = x and cn = 1 for all n:
In other words, the function f(x) can be written as a power series.
with

12. Example 1 Create new power series for other functions through: and
sum, difference, multiplication, division, composition
and
differentiation and integration
Example 1
with

13. Example 2
with

14. Consider the graphs:

15. Example 3
with

16. Need to solve for C. Set x = 0 to get:
Test endpoints???

17. Example 4
with

18. Need to solve for C. Set x = 0 to get:
Test endpoints???