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

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

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

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

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

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

TERM-081

MACLAURIN SERIES

TERM-091

MACLAURIN SERIES

TERM-101

MACLAURIN SERIES

TERM-082

)2cos(cos2

1

2

12 xx

MACLAURIN SERIES

TERM-102

MACLAURIN SERIES

TERM-091

MACLAURIN SERIES

TAYLOR AND MACLAURIN

Example:

0 !

1

n nFind the sum of the series

TERM-102

MACLAURIN SERIES

TERM-082

MACLAURIN SERIES

TERM-131

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

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

Important Maclaurin Series and Their Radii of Convergence

Example:

)1ln()( xxf

Find Maclaurin series

MACLAURIN SERIES

TERM-122

MACLAURIN SERIES

TERM-082

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