MAE101 CAL v2 chapter 8 power series

Topics :6.1 Power series and Functions 6.3 Taylor and Maclaurin series... Power series centered at a is a series of the form:... 6.1 Power series and Functions... Definition The radius o

(Page 531-581, Calculus Volume 2)Power Series

Dr Tran Quoc Duy

Email: duytq4@fpt.edu.vn

Topics :

6.1 Power series and Functions

6.3 Taylor and Maclaurin series

( )k

k k

c x



=



Power series is a series of the form:

where x is a variable, and ck are constants called the

coefficients of the series.

Power series centered at a is a series of the form:

6.1 Power series and Functions

Theorem

Let be a power series centered at a There are

only three possibilities:

• The series converges only when x= a

• The series converges for all x

• There is a positive number R such that the series

converges if |x-a|<R and diverges if |x-a|>R

k

k

k x a c

Power series

Definition

The radius of convergence of the power series

and diverges for |x – a| > R.

Interval of convergence of the power series is the interval

that consists of all x for which the series converges.

0

( )k

k k

1lim

c

k k

k

k k

)1(

x

Alternating series converges

→ the interval of convergence is [-1,1)

( 4

) 1 (

n

n n

n

x n

We have known:

→ The function 1/(1-x) can be expressed as a power series

on interval of convergence

Express 1/(2+x) as a power series and find the interval of

convergence

Solution:

Differentiating and Integrating Power Series

convergence

Example

Find the representation of ln(1+x) from that of 1/(1−x).

1

11

0

+

++

1)

(1

0

+

−+

)

(

2 0

0

0 0 0



−+

dt dt

t t

k

x

k x

)1

Definition

Taylor series of the differentiable function f(x):R →R at x=a is:

When a = 0, it is called Maclaurin series

The nth-degree Taylor polynomial of f(x) at a:

f a

x a k

k n

(

!

)0(

k

k

k

x k

If f(x) = e x , then f (k) (x) = e x → Taylor series at x=0 is:

!3

!21

3 2

++

++

Theorem

Let f(x):R→R be a function differentiable at x = a

Let R n (x) = f(x) - T n (x), where T n (x) is the nth degree Taylor

series

If limn→∞R n (x)=0 for |x − a| < R then f(x) = sum of its Taylor

series for |x − a| < R, meaning

) ( lim

) ( '

' )

)(

( ' )

( + − + − 2 +

= f a f a x a f a x a

Taylor’s Fomular

If f(x) has n+1 derivatives on an interval I that contains the

number a, then for x in I there is a number a < z < x such that :

n n

1 )

Example

+

+

++

k

x e

k

k x

!3

!2

1

!

3 2

0

+

+

−+

x k

x x

k

k k

!6

!4

!2

1)!

2(

)1

()

cos(

6 4

−+

−

=+

x x

k

x x

k

k k

!7

!5

!3)!

12

(

)1

()

sin(

7 5

3

0

1 2

1

11

0

+

++

43

21

)1

()

1

ln(

4 3

2

0

1

+

−+

−

=+

x x

k

x x

k

k k

3 4

!

)3

(

2

k

k x

k

x x

e x

!

)3(

k

k k

!

)3(

k

k k

k x

) 2 )(

1 (

! 2

) 1

( 1

) 1

( + = + + − 2 + − − 3 +

→ x p px p p x p p p x

Theorem

When |x| < 1, then (1+x) n (n is any real number) can be

represented by the binomial series.

k k

n

x k

k n

n n

2 )(

1

( )

1

(

! 3

) 2 )(

1 (

! 2

) 1

(

1 + + − 2 + − − 3 +

= nx n n x n n n x

Example

Find the binomial series of 9 − x

2 / 1

9

1

39

19

13

9

n

n

x n

x

3888216

6

3

3 2