1. Polynomial with infinit-degree
POWER SERIES
Example
Polynomial of degree n
Power Series
(Polynomial with infinit-degree)
Example
Power Series
Polynomial with infinit-degree

2. at each x  infinite series
POWER SERIES
Example
Polynomial of degree n
Power Series
(Polynomial with infinite-degree)
Example
Power Series
at each x  infinite series

3. POWER SERIES Example Polynomial of degree n Power Series Example
(Polynomial with infinit-degree)
Example
domain is the set of all x for which the series converges.

4. POWER SERIES Polynomial of degree n Example Power Series Example
Polynomial centered at a or a polynomial about a
Example
Polynomial centered at 1
Polynomial centered at 0
Power Series
centered at a
Example
Remark:
1) Maclaurin Series
centered at 0

5. POWER SERIES Theorem: Theorem: 1 1 2 2 3 3 Example Definition: 3
centered at a
Theorem:
Theorem:
there are only three possibilities:
there are only three possibilities:
The series converges only when 1 1
The series converges for all 2 2
There is a positive number R such that the series
There is a positive number R such that 3 3
converges if
converges if
diverges if
Example
Definition:
The number R in case is called the radius of convergence of the power series. 3

6. POWER SERIES Definition: 3 Theorem: Theorem: 1 1 2 2 3 3 Example
The number R in case is called the radius of convergence of the power series. 3
Theorem:
Theorem:
there are only three possibilities:
there are only three possibilities:
Radius of convergence
The series converges only when 1 1
The series converges for all 2 2
There is a positive number R such that the series 3
converges if 3
diverges if
Example

7. POWER SERIES Theorem: 1 1 2 2 3 3 Example Example
How to find R = radius of convergence
there are only three possibilities:
Radius of convergence
Find: 1 1
Find: 2
(L is a function of x only) 2 3
Use ratio test: 3
Example
Example
Find the radius of convergence
Find the radius of convergence

8. POWER SERIES
Final-082

9. POWER SERIES
Final-102

10. POWER SERIES Theorem: Theorem: 1 1 2 2 3 3 Example Remark: 3 divg
there are only three possibilities:
there are only three possibilities:
Radius of convergence
The series converges only when 1 1
The series converges for all 2 2
There is a positive number R such that the series 3 3
converges if
diverges if
Example
Remark:
case say nothing about the endpoints 3
converges if
diverges if
divg
convg
divg

11. POWER SERIES Definition: Definition: 3 Theorem: Theorem: 1 1 2 2 3 3
The interval of convergence is the interval that consists of all values of x for which the series converges.
The number R in case is called the radius of convergence of the power series. 3
Theorem:
Theorem:
there are only three possibilities:
Interval of convergence
there are only three possibilities:
Radius of convergence 1 1 2 2 3 3
Remark:
Example
endpoint of the interval, that is, ,
anything can happen—the series might converge at one or both endpoints or it might diverge at both endpoints.
Remark:
Thus in case there are four possibilities for the interval of convergence: 3

12. POWER SERIES 1 2 3 1 2 How to find R = radius of convergence Find:
(L is a function of x only)
Use ratio test: 3
How to find interval of convergence
Find R: 1
Study convg at endpoints: a+R and a-R 2

13. POWER SERIES convg divg Example Example Example Example
(Bessel function of order 0)
Find the interval of convergence
Find the interval of convergence
Example
Example
Find the interval of convergence
Find the interval of convergence

14. POWER SERIES
Final-101

15. POWER SERIES
Final-092

16. POWER SERIES
Final-081

17. POWER SERIES
Final-081

18. POWER SERIES
Final-102

19. POWER SERIES
Final-092

20. POWER SERIES
Final-092

21. POWER SERIES
Final-082

22. DIFFERENTIATION Theorem: 1 2 Example: POWER SERIES
has radius of convergence 2
radius of convergence of is R
Example:
radius of convergence of is 1
radius of convergence of is ??

23. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
INTEGRATION
Theorem: 1
has radius of convergence 2
radius of convergence of is R
Example:
radius of convergence of is 1
radius of convergence of is ??
Remark:
Although Theorem says that the radius of convergence remains the same
when a power series is differentiated, this does not mean that the interval of convergence remains the same. It may happen that the original series converges at an endpoint, whereas the differentiated series diverges there. (also for integration)

24. Operations on Power Series Operations on Power Series
Add:
Intersection of interval of convergence
Subtrac:
Multiplicat:
Example:
Find the first 3 terms of