1. Ch 9.4 Radius of Convergence Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy

6. Importance of Convergence

8. This equation is an identity only for x in (-1,1). Convergence matters!

9. What You’ll Learn When does a power series converge? If a power series converges, over what interval of x? –The n th Term Test for Divergence There are 4 tests for convergence: –The Comparison Test –Absolute Convergence Test for Alternating Series –The Ratio Test –Endpoint Convergence

10. Note: R is the radius of convergence. This result is useless since the series is only accurate at the center.

11. Cases of Convergence Case 1: Converges over a radius of 1 or -1 < x < 1. Case 2: Converges for all values of x. Case 3: Converges only at center or x = a. We will see an example for homework.

12. The nth Term Test for Divergence for non-negative terms only First we need to rule out divergence:

13. Direct Comparison Test

14. Example of Comparison Test

16. Note: This test is for alternating series.

17. Example of Absolute Convergence

24. If L = 1, Ratio Test is Inconclusive

26. Endpoint Convergence

28. Summing a Telescoping Series