1. Power Series

2. A power series in x (or centered at 0) is a series of the following form:

3. Accepted bad Convention When writing the power series in the form shown on the right, we follow the inaccurate convention that the expression x 0 should be replaced by 1, when x = 0.

4. A power series in x-c (or centered at c) is a series of the following form:

5. Accepted bad Convention When writing the power series in the form shown on the right, we follow the inaccurate convention that the expression (x-c) 0 should be replaced by 1, when x = c.

6. Examples I Geometric series are power series

7. Example (1) A power series centered at 0 and of interval of convergence (-1,1)

8. Example (2) A power series centered at 0 and and of interval of convergence (-5,5)

9. Example (3) A power series centered at 2 and and of interval of convergence (-3,7)

10. Convergence of power series Investigating the convergence of a power series is determining for which values of x the series converges and for which values it diverges.

11. Every power series converges at least at one point; its center Obviously the power series converges for x = c. To determine the other values of x, for which the series converges, we often use the ratio test

12. Going back to the previous examples

13. Examples II Convergence of other power series

14. Example (1)

16. Convergence at the end-points of the interval

17. Conclusion The series converges on the interval [- ½, ½ )

18. Example (2)

20. Convergence at the end-points of the interval

21. Conclusion The series converges on the interval (- 1, 5 )

22. Example (3)

24. Example (4)

26. Theorem A power series of the form Is either absolutely convergent everywhere, only at its center or on some interval about its center.

27. The three cases Case (1): we say that the series is absolutely convergent on R or on ( -∞, ∞) and that the radius of convergence is ∞ Case (2): we say that the series is convergent at x = c and divergent everywhere else, and that the radius of convergence is 0. Case (3): The series is absolutely convergent at an interval of the form ( c-r,c+r), for some positive number r, and divergent on (-∞,c-r)U(c+r, ∞). In this case we say that the interval of convergence is equal to ( c-r,c+r) and the radius of convergence is equal to r. We investigate separately the convergence of the series at each of the end points of the interval

28. Homework Determine the interval & radius of convergence of the given series

29. Hints