Taylor and Maclaurin Series

Lesson 9.10

Convergent Power Series Form

• Consider representing f(x) by a power series

• For all x in open interval I• Containing c

• Then

( )n

nf x a x c

( ) ( )

!

n

n

f ca

n

2"( )( ) ( ) '( )( ) ( ) ...

2!

f cf x f c f c x c x c

Taylor Series

• If a function f(x) has derivatives of all orders at x = c, then the series

is called the Taylor series for f(x) at c.

• If c = 0, the series is the Maclaurin series for f .

( )2

0

( ) "( )( ) ( ) '( )( ) ( ) ...

! 2!

nn

n

f c f cx c f c f c x c x c

n

Taylor Series

• This is an extension of the Taylor polynomials from section 9.7

• We said for f(x) = sin x, Taylor Polynomial of degree 7

2 3

7

4 5 6 7

sin ( ) 0 02! 3!

0 04! 5! 6! 7!

x xx M x x

x x x x

Guidelines for Finding Taylor Series

1. Differentiate f(x) several times

• Evaluate each derivative at c

2. Use the sequence to form the Taylor coefficients

• Determine the interval of convergence

3. Within this interval of convergence, determine whether or not the series converges to f(x)

( )( ), "( ), '''( ),..., ( )nf c f c f c f c

( ) ( )

!

n

n

f ca

n

Series for Composite Function

• What about when f(x) = cos(x2)?

• Note the series for cos x

• Now substitute x2 in for the x's

2 4 6

cos 1 ...2! 4! 6!

x x xx

4 8 122cos 1 ...

2! 4! 6!

x x xx

Binomial Series

• Consider the function• This produces the binomial series

• We seek a Maclaurin series for this function• Generate the successive derivatives• Determine• Now create the series using the pattern

( ) 1k

f x x

( ) (0) ?nf

2"( )( ) (0) '(0) ...

2!

f cf x f f x x

Binomial Series

•

• We note that

• Thus Ratio Test tells us radius of convergence R = 1

• Series converges to some function in interval-1 < x < 1

2( 1) ( 1) ... ( 1)( ) (1 ) 1 ...

2 !

nk k k x k k k n x

f x x kn

1lim 1n

nn

a

a

Combining Power Series

• Consider

• We know

• So we could multiply and collect like terms

( ) arctanxf x e x

2 3

3 5

1 ...2! 3!

arctan ...3 5

x x xe x

and

x xx x

2 31arctan ...

6xe x x x x

Assignment

• Lesson 9.10

• Page 685

• 1 – 29 odd