Power Series

Radii and Intervals of Convergence

First some examples

Consider the following example series:

0

2

!

k

k k

• What does our intuition tell us about the convergence or divergence of this series?

• What test should we use to confirm our intuition?

Power Series

Now we consider a whole family of similar series:

0

2

!

k

k k

0

1

!

k

k k

0

4

!

k

k k

0

12!

k

k k

0

214

!

k

k k

• What about the convergence or divergence of these series?

• What test should we use to confirm our intuition?

We should use the ratio test; furthermore, we can use the similarity between the series to test them all at once.

0 !

k

k

x

k

Power Series

Now we consider a whole family of similar series:

0

2

!

k

k k

0

1

!

k

k k

0

4

!

k

k k

0

12!

k

k k

0

214

!

k

k k

• What about the convergence or divergence of these series?

• What test should we use to confirm our intuition?

But remember that the ratio test applies only to series with positive terms---we will be testing for absolute convergence!

0 !

k

k

x

k

How does it go? We start by setting up the appropriate limit.

1!

lim1 !

k

kk

x k

kx

1

1 !lim

!

k

kk

x

k

x

k

lim

1k

x

k

0

Since the limit is 0 which is less than 1, the ratio test tells us that the series

converges absolutely for all values of x.

0 !

k

k

x

k

Why the absolute values?Why on the x’s and not elsewhere?

The series is an example of a power series.0 !

k

k

x

k

What are Power Series?It’s convenient to think of a power series as an infinite polynomial:

Polynomials:

Power Series:

2 3 4

0

3 3 3 3 1 31

3! 5! 7! 9! 2 1 !

k k

k

x x x x x

k

2 311 ( 1) 3( 1) ( 1)4x x x

2 52 3 12x x x

2 3 4

0

1 2 3 4 5 ( 1) k

k

x x x x k x

In general. . .

Definition: A power series is a (family of) series of the form

00

( ) .nn

n

a x x

In this case, we say that the power series is based at x0 or that it is centered at x0.

What can we say about convergence of power series? A great deal, actually.

Checking for Convergence

I should use the ratio test. It is the test of choice when testing for convergence of power series!

Checking for Convergence

Checking on the convergence of

2 3 4

0

1 2 3 4 5 ( 1) k

k

x x x x k x

We start by setting up the appropriate limit.

x1

( 2)lim

( 1)

k

kk

k x

k x

( 2)

lim( 1)k

k x

k

The ratio test guarantees convergence provided that this limit is less than 1. That is, when |x| < 1.

0 11

Checking for Convergence

Checking on the convergence of

2 3 4

0

1 2 3 4 5 ( 1) k

k

x x x x k x

We start by setting up the appropriate limit.

x1

( 2)lim

( 1)

k

kk

k x

k x

( 2)

lim( 1)k

k x

k

The ratio test is inconclusive if |x| = 1. (x = 1.)We have to test these separately!0 11

Checking for Convergence

Checking on the convergence of

2 3 4

0

1 2 3 4 5 ( 1) k

k

x x x x k x

We start by setting up the appropriate limit.

x1

( 2)lim

( 1)

k

kk

k x

k x

( 2)

lim( 1)k

k x

k

The ratio test is inconclusive if |x| = 1. (x = 1.)We have to test these separately!0 11

One More Example:

Determine where the series

We start by setting up the ratio test limit.

13

2( 1) 1 !lim

3

2 1 !

k

kk

x

k

x

k

13 (2 1)!

lim(2 3)!3

k

kk

x k

kx

0

1 3

2 1converg .

!es

k k

k

x

k

3

lim2 2 2 3k

x

k k

3 1 2 3 4 2 1 2 (2 1)lim

1 2 3 4 2 1 2 (2 1) 2 2 2 3k

x k k k

k k k k k

Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely no matter what x is.

0

3

What about the convergence of

We start by setting up the ratio test limit.

13

2( 1) 1 !lim

3

2 1 !

k

kk

x

k

x

k

13 (2 1)!

lim(2 3)!3

k

kk

x k

kx

2 3 4

0

3 3 3 3 1 31 ?

3! 5! 7! 9! 2 1 !

k k

k

x x x x x

k

3

lim2 2 2 3k

x

k k

3 1 2 3 4 2 1 2 (2 1)lim

1 2 3 4 2 1 2 (2 1) 2 2 2 3k

x k k k

k k k k k

Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely no matter what x is.

0

3

Now you work out the convergence of

2 3 4

0

3 3 3 3 1 31

3 5 7 9 2 1

k k

k

x x x x x

k

Don’t forget those absolute values!

Now you work out the convergence of

We start by setting up the ratio test limit.

13

2( 1) 1lim

3

2 1

k

kk

x

k

x

k

13 (2 1)

lim(2 3)3

k

kk

x k

kx

2 3 4

0

3 3 3 3 1 31

3 5 7 9 2 1

k k

k

x x x x x

k

What does this tell us?

(2 1)3 lim

(2 3)k

kx

k

3x

• The power series converges absolutely when | x+3| < 1. • The power series diverges when | x+3 | > 1. • The ratio test is inconclusive for x = -4 and x = -2. (Test

these separately… what happens?)

Convergence of Power SeriesWhat patterns can we see? What conclusions can we draw?

When we apply the ratio test, the limit will always be either 0 or some positive number times | x – x0 |. (Actually, it could be , too. What would this mean?)

• If the limit is 0, the ratio test tells us that the power series converges absolutely for all x.

• If the limit is k | x - x0 |, the ratio test tells us that the series converges absolutely when k | x - x0 | < 1. It diverges when k | x - x0 | > 1. It fails to tell us anything if k | x - x0 | = 1.

What does this tell us?

0

1 when | - | the series converges absolutely.x x

k

0

1 when | - | the series diverges.x x

k

0

1 when | - | we don't know.x x

k

Suppose that the limit given by the ratio test is 0| - | .k x x

We need to consider separately the cases when

• k | x - x0 | < 1 (the ratio test guarantees convergence),• k | x - x0 | > 1 (the ratio test guarantees divergence), and • k | x - x0 | = 1 (the ratio test is inconclusive).

This means that . . . Recall that k 0 !

Recap

0

1 when | - | the series converges absolutely.x x

k

0

1 when | - | the series diverges.x x

k

0

1 when | - | we don't know. x x

k

0x 0

1x

k0

1x k

Must test endpoints separately!

ConclusionsTheorem: If we have a power series ,

• It may converge only at x = x0.

00

( )nn

n

a x x

• It may converge for all x.

• It may converge on a finite interval centered at x=x0.

0x 0x R0x R

Radius of conv. is R.

0x

0x

Radius of convergence is 0

Radius of conv. is infinite.

Radius of Convergence vs. Interval of Convergence

The set of x values for which the power series

converges, is called the interval of convergence of the power series. The radius of this interval is called the radius of convergence of the power series.

00

( )nn

n

a x x

0

( 1) k

k

k x

For Example:

0 11

Interval of convergence is (-1,1)Radius of convergence is 1

Radius of Convergence vs. Interval of Convergence

Examples:

0

Interval of convergence is (- , )Radius of convergence is infinite

0 !

k

k

x

k

Interval of convergence is (-4 , -2]Radius of convergence is 1

0

1 3

2 1

k k

k

x

k

3 24

We start by setting up the ratio test limit.

0

2 2

ln( 3)converges.

kk

k

x

k

132 2 lim

14

k

kx

k

112 2

ln( 4)lim

2 2

ln( 3)

kk

kkk

x

k

x

k

112 2 ln( 3)lim

ln( 4)2 2

kk

kkk

x k

kx

ln( 3)2 2 lim

ln( 4)k

kx

k

42 2 lim 2 2

3k

kx x

k

The ratio test guarantees convergence provided that 2|x+2| < 1. That is, if |x+2| < 1/2

Where does the series diverge? Where does the

ratio test fail us?

One More Example:

Determine where the series

What about the convergence of

We start by setting up the ratio test limit.112 2

ln( 4)lim

2 2

ln( 3)

kk

kkk

x

k

x

k

112 2 ln( 3)lim

ln( 4)2 2

kk

kkk

x k

kx

ln( 3)2 2 lim

ln( 4)k

kx

k

132 2 lim

14

k

kx

k

4

2 2 lim 2 23k

kx x

k

2 325

2Check the endpoints!

0

2 2 ?

ln( 3)

kk

k

x

k

What about the convergence of

We start by setting up the ratio test limit.112 2

ln( 4)lim

2 2

ln( 3)

kk

kkk

x

k

x

k

112 2 ln( 3)lim

ln( 4)2 2

kk

kkk

x k

kx

ln( 3)2 2 lim

ln( 4)k

kx

k

132 2 lim

14

k

kx

k

4

2 2 lim 2 23k

kx x

k

2 325

2

c.c.Radius of Convergence is ½.Interval of convergence is [-5/2, -3/2)

0

2 2 ?

ln( 3)

kk

k

x

k