maclaurin series

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MACLAURIN SERIES 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) Finding limit Finding a sum of a series (not only geometric, telescoping) dx e x 2 2 0 1 lim x x e x x

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MACLAURIN SERIES. 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) Finding limit - PowerPoint PPT Presentation

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Page 1: MACLAURIN SERIES

MACLAURIN SERIES

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)

Finding limit

Finding a sum of a series (not only geometric, telescoping)

dxex2

20

1lim

x

xex

x

Page 2: MACLAURIN SERIES

Example: xexf )(

0n

nn

x xce 55

44

33

2210 xcxcxcxcxcc

Maclaurin series ( center is 0 )

Example:

xxf sin)( Find Maclaurin series

MACLAURIN SERIES

xxf sin)(

xxf cos)()1(

xxf sin)()2(

xxf cos)()3(

xxf sin)()4(

0)0( f

1)0()1( f

0)0()2( f

1)0()3( f

0)0()4( f

!

)0()(

n

nfnc

Page 3: MACLAURIN SERIES

Example:

xxf cos)( Find Maclaurin series

MACLAURIN SERIES

xxdx

dcossin

Example:

xxf

1

1)(

Find Maclaurin series

Example:

21

1)(x

xf

Find Maclaurin series

0

22

)(1

1

n

nxx

2by each replace xx

0

2)1(

n

nn x

Example:

xxf 1tan)(

Find Maclaurin series

integrate

0

)1(2

1 2

1tan

n

dxxn n

x

dxx

0

12)1(

12

nn

xn n

Example:

xxf

1

1)(

Find Maclaurin series

0

)(1

1

n

nxx

xx by each replace

0

)1(

n

nn x

Page 4: MACLAURIN SERIES

Important Maclaurin Series and Their Radii of Convergence

MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620

MACLAURIN SERIES

How to memorize them

Page 5: MACLAURIN SERIES

Important Maclaurin Series and Their Radii of Convergence

MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620

MACLAURIN SERIES

Denominator is n!

even, odd

Denominator is nodd

Page 6: MACLAURIN SERIES

MACLAURIN SERIES

Maclaurin series ( center is 0 ) !

)0()(

n

nfnc

How to find a Maclaurin Series of a function

Use the formula Use the known functions

1) Replace each x2) Diff 3) integrate3) Find a product between two

Page 7: MACLAURIN SERIES

TERM-081

MACLAURIN SERIES

Page 8: MACLAURIN SERIES

TERM-091

MACLAURIN SERIES

Page 9: MACLAURIN SERIES

TERM-101

MACLAURIN SERIES

Page 10: MACLAURIN SERIES

TERM-082

)2cos(cos2

1

2

12 xx

MACLAURIN SERIES

Page 11: MACLAURIN SERIES

TERM-102

MACLAURIN SERIES

Page 12: MACLAURIN SERIES

TERM-091

MACLAURIN SERIES

Page 13: MACLAURIN SERIES

TAYLOR AND MACLAURIN

Example:

0 !

1

n nFind the sum of the series

Page 14: MACLAURIN SERIES

TERM-102

MACLAURIN SERIES

Page 15: MACLAURIN SERIES

TERM-082

MACLAURIN SERIES

Page 16: MACLAURIN SERIES

TERM-131

MACLAURIN SERIES

Page 17: MACLAURIN SERIES

Leibniz’s formula:

Example: Find the sum

0

121

12)1()(tan

n

nn

n

xx

753)(tan

7531 xxx

xx

0 12

)1(

n

n

n

7

1

5

1

3

11

MACLAURIN SERIES

Page 18: MACLAURIN SERIES

Important Maclaurin Series and Their Radii of Convergence

MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620

MACLAURIN SERIES

Denominator is n!

even, odd

Denominator is nodd

MAC

Page 19: MACLAURIN SERIES

Important Maclaurin Series and Their Radii of Convergence

Example:

)1ln()( xxf

Find Maclaurin series

MACLAURIN SERIES

Page 20: MACLAURIN SERIES

TERM-122

MACLAURIN SERIES

Page 21: MACLAURIN SERIES

TERM-082

MACLAURIN SERIES

Page 22: MACLAURIN SERIES

Important Maclaurin Series and Their Radii of Convergence

MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620

MACLAURIN SERIES

Denominator is n!

even, odd

Denominator is nodd