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Sec 11.9 & 11.10: REPRESENTATIONS OF FUNCTIONS AS POWER SERIES

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Presentation on theme: "Sec 11.9 & 11.10: REPRESENTATIONS OF FUNCTIONS AS POWER SERIES"— Presentation transcript:

1 Sec 11.9 & 11.10: REPRESENTATIONS OF FUNCTIONS AS POWER SERIES
TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known function as a sum of infinitely many terms. Integration. (Easy to integrate polynomials) approximating functions by polynomials. Finding limit Finding a sum of a series (not only geometric, telescoping)

2 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Example: Maclaurin series ( center is 0 ) Example: Find Maclaurin series

3 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence SYLLABUS: Students must know the Maclaurin Series listed in Table I of page 743.

4 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 ) Example: Find Maclaurin series

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TERM-081

6 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091

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TERM-101

8 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-082

9 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 ) Example: Find the sum of the series

10 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102

11 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-082

12 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence SYLLABUS: Students must know the Maclaurin Series listed in Table I of page 743.

13 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
DEF: Example: Example:

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binomial series. NOTE:

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TERM-101 binomial series.

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TERM-092 binomial series.

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Theorem: 1 differentiable on has radius of convergence 2 continuous on DIFFERENTIATION DIFFERENTIATION Theorem: Theorem: 1 1 has radius of convergence has radius of convergence 2 2 radius of convergence of is R radius of convergence of is R

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DIFFERENTIATION Theorem: 1 has radius of convergence 2 radius of convergence of is R 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.

19 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
DIFFERENTIATION 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 ??

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TERM-102

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TERM-091

22 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
INTEGRATION Theorem: 1 has radius of convergence 2 radius of convergence of is R 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.

23 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence Example: Find Maclaurin series

24 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence Example: Find Maclaurin series

25 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102

26 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-111

27 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-101

28 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-082

29 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence SYLLABUS: Students must know the Maclaurin Series listed in Table I of page 743.

30 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 ) Taylor series ( center is a )

31 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091

32 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-092

33 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-082

34 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102

35 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence SYLLABUS: Students must know the Maclaurin Series listed in Table I of page 743.

36 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-092

37 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-081

38 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-101

39 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-081

40 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Taylor series ( center is a ) SYLLABUS: (Theorems 8 & 9 are not Included.) DEF: nth-degree Taylor polynomial of f at a.

41 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Taylor series ( center is a ) DEF: nth-degree Taylor polynomial of f at a. DEF: Remainder

42 Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
Taylor series ( center is a ) DEF: nth-degree Taylor polynomial of f at a. DEF: Remainder Example:

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