Def: The power series centered at x = a:

1

x is the variable and the c’s are constants (coefficients)

Lecture 29 – Power Series

)()()( 2210

0

axcaxccaxcn

nn

0,!

1a

ncn

!

!3!211)0(

!

1 32

0 k

xxxxx

n

k

n

n

2,

2

11

acn

n

n

1

2

01 2

)2()1(

8

)2(

4

2

2

1)2(

2

1k

kk

n

nn

n xxxx

2

For any power series, exactly one of the following is true:

.3

.2

.1 x allfor converges

ax only for converges

) (in allfor )absolutelyconverges( RR,x

) ( outside allfor diverges & RR,x

divergeor convergemay ,or at & R x Rx

Example 1 – Radius and Interval of Convergence

3

n

n

n x

n

n

x !

)!1(lim

1

Ratio Test:

!

!3!211

!

32

0 k

xxxx

n

x k

n

n

Series converges for

Example 2 – Radius and Interval of Convergence

4

n

n

n xn

xn

)10(!

)10(! )1(lim

1

Ratio Test:

k

n

n xkxxxn )10(! )10(!2)10(10! 2

1

Series converges for

Example 3 – Radius and Interval of Convergence

5

n

n

n x

n

n

x

)2(1

)2(lim

1

Ratio Test:

k

xxxx

n

x k

n

n )2(

3

)2(

2

)2(2

2 32

1

Example 3 – continued (testing endpoints)

6

R

:Interval ,

k

xxxx

n

x k

n

n )2(

3

)2(

2

)2(2

2 32

1

:2

1x

:2

1x

Example 4 – Radius and Interval of Convergence

7

n

n

n

x

3

1lim

Root Test:

k

k

n

n

nn

n xxxxx

3

)1(

9

)1(

3

11

3

1

3

1 2

00

Example 4 – continued (testing endpoints)

8

00 3

1

3

1

n

n

nn

n xxR :Interval ,

:4x

:2x

Example 5 – Radius and Interval of Convergence

9

Geometric Series:

1

2

01 2

)1()1(

8

)2(

4

2

2

1

2

21k

kk

nn

nn xxxx

001

2

21

2

1

2

21

nn

nn

nn

nn xx

Example 5 – continued – what is the converging value?

10

Geometric Series:

1

2

01 2

)1()1(

8

)2(

4

2

2

1

2

21k

kk

nn

nn xxxx

0 2

2

2

1

n

nx

The geometric series:

.1 when 1

2

|r|r

aarara

11

Lecture 30 – More Power Series

As a power series with a = 1, r = x and cn = 1 for all n:

.1 when 1

11 2

|x|

xxx

In other words, the function f(x) can be written as a power series.

0

1

1)(

n

nxx

xf

1 ,1 :Interval with 1R

Create new power series for other functions through:sum, difference, multiplication, division, composition

anddifferentiation and integration

12

0

1

1)(

n

nxx

xf 1,1 :Interval with 1R

)3()( xfxg

Example 1

13

1

)(2x

xxh

0

1

1)(

n

nxx

xf 1,1 :Interval with 1RExample 2

14

Consider the graphs:

1

)(2x

xxh

1,1 :Interval 753 xxxx

xy 1

33 xxy

535 xxxy

7537 xxxxy 11

)(xh1

15

tan)( 1 xxf

0

1

1

n

nxx

1,1 :Interval with 1RExample 3

16

tan 1 x

Need to solve for C. Set x = 0 to get:

12

)1(0

12

n

nn

n

x tan 1 x

:1x :1x

Test endpoints???

17

)1ln()( xxf

0

1

1

n

nxx

1,1 :Interval with 1RExample 4

18

)1ln( x

Need to solve for C. Set x = 0 to get:

1

)1(0

1

n

nn

n

x )1ln( x

:1x :1x

Test endpoints???