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Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When we easily see that it diverges when x=4 and converges when x=1. Thus the power series converges for 1 ( 3) n n x n 1 ( 3) lim lim | 3| 1 n n n n a nx x a n | 3| 1 x | 3| 1. x | 3| 1, x 2 4. x

Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

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Page 1: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. For what values of x is the power series

convergent? Sol. By ratio test,

the power series absolutely converges when

and diverges when When we easily

see that it diverges when x=4 and converges when x=1. Thus

the power series converges for

1

( 3)n

n

x

n

1 ( 3)lim lim | 3 |

1n

n nn

a n xx

a n

| 3 | 1x | 3 | 1.x | 3 | 1,x

2 4.x

Page 2: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. Find the domain of the Bessel function defined by

Sol. By ratio test,

the power series absolutely converges for all x. In other words,

The domain of the Bessel function is

2

2 20

( 1)( )

2 ( !)

n n

nn

xJ x

n

21

2lim lim 0

4( 1)n

n nn

a x

a n

( , ).

Page 3: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Characteristic of convergence Theorem For a given power series

there are only three possibilities:

(i) The series converges only when x=a.

(ii) The series converges for all x.

(iii) There is a positive number R such that the series

converges if and diverges if The number R is called the radius of convergence of the

power series. By convention, in case (i) the radius of

convergence is R=0, and in case (ii)

0

( )nnn

c x a

.R

| |x a R | | .x a R

Page 4: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Characteristic of convergence The interval of convergence of a power series is the

interval that consists of all x for which the series converges. To find the interval of convergence, we need to determine

whether the series converges or diverges at endpoints |x-a|=R. Ex. Find the radius of convergence and interval of

convergence of the series

Sol

radius of convergence is 1/3.

At two endpoints: diverge at 5/3, converge at 7/3. (5/3,7/3]

0

( 3) ( 2)

1

n n

n

x

n

1lim | | 3 | 2 |n

nn

ax

a

3 | 2 | 1 | 2 | 1/ 3x x R

Page 5: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Radius and interval of convergence From the above example, we found that the ratio test or the

root test can be used to determine the radius of convergence. Generally, by ratio test, if then

By root test, if then Ex. Find the interval of convergence of the series

Sol. when |x|=1/e, the general does not have limit zero, so diverge. (-1/e,1/e)

2

1

1( )n n

n

nx

n

1lim | | ,n

nn

c

c

1/ .R

lim | | ,nn

nc

1/ .R

lim | | 1/nn

nc e R e

Page 6: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Representations of functions as power series

We know that the power series converges to

when –1<x<1. In other words, we can represent the function

as a power series

Ex. Express as the sum of a power series and find the interval of convergence.

Sol. Replacing x by in the last equation, we have

0

n

n

x

1

1 x

2 3

0

11 (| | 1)

1n

n

x x x x xx

21/(1 )x

2x

2 2 4 6 82 2

0

1 1( ) 1

1 1 ( )n

n

x x x x xx x

Page 7: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. Find a power series representation for Sol.

The series converges when |-x/2|<1, that is |x|<2. So the

interval of convergence is (-2,2).

Question: find a power series representation for

Sol.

1 1 1 1 1

2 2 1 / 2 2 1 ( / 2)x x x

10 0

1 1 ( 1)( )

2 2 2 2

nn n

nn n

xx

x

1/( 2).x

3 /( 2).x x 3

3 3 31 1

0 0

1 ( 1) ( 1)

2 2 2 2

n nn n

n nn n

xx x x x

x x

Page 8: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Differentiation and integration Theorem If the power series has radius of

convergence R>0, then the sum function

is differentiable on the interval and

(i)

(ii)

The above two series have same radius of convergence R.

( )nnc x a

20 1 2

0

( ) ( ) ( ) ( )nn

n

f x c x a c c x a c x a

( , )a R a R 1

1 21

( ) 2 ( ) ( )nnn

f x c c x a nc x a

2 1

0 10

( ) ( )( ) ( )

2 1

n

nn

x a x af x dx C c x a c C c

n

Page 9: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example The above formula are called term-by-term

differentiation and integration. Ex. Express as a power series and find the radius

of convergence. Sol. Differentiating gives

By the theorem, the radius of convergence is same as the

original series, namely, R=1.

21/(1 )x

0

1

1n

n

xx

1 22

1

11 2 3

(1 )n

n

nx x xx

Page 10: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. Find a power series representation of Sol.

( ) arctan .f x x

2 2 2 4 62

0 0

1( ) ( 1) 1

1n n n

n n

x x x x xx

2 1

20

1arctan ( 1)

1 2 1

nn

n

xx dx C

x n

2 1

0

arctan 0 0 0 arctan ( 1)2 1

nn

n

xC x

n

Page 11: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. Find a power series representation for

and its radius of convergence.

Sol.

( ) ln(1 )f x x

2 31ln(1 ) (1 )

1x dx x x x dx

x

2

12

n

n

x xC x C

n

1

ln(1 0) 0 0 ln(1 ) .n

n

xC x

n

1

1ln 2.

2nn n

1.R

Page 12: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Taylor series Theorem If f has a power series representation

(expansion) at a, that is, if

Then its coefficients are given by the formula

This is called the Taylor series of f at a (or about a)

( ) ( )

!

n

n

f ac

n

0

( ) ( ) | |nn

n

f x c x a x a R

( )2

0

( ) ( ) ( )( ) ( ) ( ) ( ) ( )

! 1! 2!

nn

n

f a f a f af x x a f a x a x a

n

Page 13: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Maclaurin series The Taylor series of f at a=0 is called Maclaurin series

Ex. Find the Maclaurin series of the function and its radius of convergence.

( ) xf x e

2 3

0

1! 1! 2! 3!

nx

n

x x x xe x

n

( )2

0

(0) (0) (0)( ) (0)

! 1! 2!

nn

n

f f ff x x f x x

n

Page 14: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Maclaurin series Ex. Find the Maclaurin series for sinx. Sol.

So the Maclaurin series is

3 5 2 1

0

( 1)

3! 5! (2 1)!

n n

n

x x xx

n

( ) ( )( ) sin( ) (0) sin2 2

n nn nf x x f

Page 15: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Important Maclaurin series Important Maclaurin series and their convergence interval

2 3

0

1 ( , )! 1! 2! 3!

nx

n

x x x xe

n

2 1 3 5 7

0

( 1)sin ( , )

(2 1)! 3! 5! 7!

n n

n

x x x xx x

n

2 3

0

11 ( 1,1)

1n

n

x x x xx

2 2 4 6

0

( 1)cos 1 ( , )

(2 )! 2! 4! 6!

n n

n

x x x xx

n

Page 16: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Example Ex. Find the Maclaurin series of Sol.

( ) 1 .f x x

1/ 2 1/ 21( ) ( 1) ( ) ( 1) ,

2f x x f x x

3/ 2 5/ 21 3( ) ( 1) , ( ) ( 1) ,

4 8f x x f x x

( ) 1 (2 1) / 23 5 (2 3)( ) ( 1) ( 1)

2n n n

n

nf x x

( ) 1 1(2 3)!! (2 3)!!(0) ( 1) , ( 1)

2 2 !n n n

nn n

n nf a

n

1

2

(2 3)!!1 1 ( 1) .

2 2 !n n

nn

x nx x

n

Page 17: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Multiplication of power series Ex. Find the first 3 terms in the Maclaurin series for

Sol I. Find and the Maclaurin series is found.

Sol II. Multiplying the Maclaurin series of and sinx

collecting terms:

sinxe x

(0), (0), (0)f f f

xe

2 3 3

sin (1 )( )2 6 6

x x x xe x x x

32

3

xx x

Page 18: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Division of power series Ex. Find the first 3 terms in the Maclaurin series for

Sol I. Find and the Maclaurin series is found.

Sol II. Use long division

tan x

(0), (0), (0)f f f

3 5

3 5

2 4

26 120tan3 15

12 24

x xx x x

x xx x

Page 19: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Application of power series Ex. Find

Sol. Let then s=S(-1/2).

To find S(x), we rewrite it as

2

0

1( 1) .

2n

nn

n ns

2

0

( ) ( 1) ,n

n

S x n n x

1

1 0

( ) ( 1) ( ) 1/(1 ),n n

n n

S x x n n x x xT x x

1 2

0 01

( ) /(1 )x x n

n

T x dx dx x x x

2

3

2( )

1 (1 )

xT x

x x

3

1 2 2 10.

2 (3 / 2) 3 27s

Page 20: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Exercise Ex. Find

Sol.

2

0

1.

3nn

n ns

2

0

( ) ( 1) ,n

n

S x n n x

2 2 2

2 0

( ) ( 1) ( ) 1/(1 ),n n

n n

S x x n n x x x T x x

2

0 02

( ) /(1 )x x n

n

T x dx dx x x x

9

.4

s

Page 21: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Application of power series Ex. Find by the Maclaurin series expansion.

Sol.

where in the last limit, we have used the fact that power

series are continuous functions.

20

1lim

x

x

e x

x

2 3

2 20 0

(1 ) 11 2! 3!lim limx

x x

x xx xe x

x x

2 3

2

20 0

1 12! 3!lim lim ,2 3! 4! 2x x

x xx x

x

Page 22: Example Ex. For what values of x is the power series convergent? Sol. By ratio test, the power series absolutely converges when and diverges when When

Homework 25 Section 11.8: 10, 17, 24, 35

Section 11.9: 12, 18, 25, 38, 39

Section 11.10: 41, 47, 48