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

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 says that the series converges provided that this limit is less than 1. That is, when |x|<1.

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 for all x.

0

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 !

Recapping

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

0x•It may converge for all x.

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

Radius of convergence is 0

Radius of conv. is infinite.

0x 0x R0x R

Radius of conv. is R.

ConclusionsTheorem: If we have a power series ,

• It may converge only at x=x0.

00

( )nn

n

a x x

0x•It may converge for all x.

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

0x 0x R0x R