1. MTH 324
Lecture # 23
Power Series

2. Previous Lecture’s Review
Derivatives of an analytic function
Cauchy’s inequality
Liouville’s theorem
Fundamental theorem of Algebra
Morera’s theorem 2

3. Review of real power series Sequence and series of complex numbers
Lecture’s Outline
Review of real power series
Sequence and series of complex numbers
Complex power series
Operations on series 3

4. Review of real sequence and series:4

5. 5

6. 6

7. Sequence: 7

8. Criterion for convergence:
Example: 8

9. Solution: 9

10. 10

11. Series:
Remark: 11

12. Geometric series :
Example: 12

13. 13

14. 14

15. Example:
Solution: 15

16. Theorem: (Necessary condition for convergence)
Proof: 16

17. Divergence Test
Example: 17

18. Absolute and conditional convergence18

19. Tests for convergence
Ratio test: 19

20. Root test: 20

21. Circle of convergence:
Power series:
Circle of convergence: 21

22. Remark: 22

23. Example:
Solution: 23

24. 24

25. Example:
Solution: 25

26. 26

27. 27

28. Theorem:
Proof: 28

29. 29

30. 30

31. References
A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan.
Complex variables and applications by James Brown and Ruel Churchill