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MTH 324 Lecture # 23 Power Series.

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Presentation on theme: "MTH 324 Lecture # 23 Power Series."— Presentation transcript:

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 convergence
18

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


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