Power Series as Functions. Defining our Function In general, we can say that That is, if the series...

Power Series as Functions

Defining our Function

00

( ) ( )nn

n

f x a x x

In general, we can say that

That is, if the series converges at x = a,

then is a number and we can

define 00

( ) ( ) .nn

n

f a a a x

00

( )nn

n

a a x

The power series is a

function on its interval of convergence.

00

( )nn

n

a x x

Graphical Example

• We saw this idea “in action” last time when we looked at the graph of some partial sums of a power series.

• This graph shows the 30th, 35th, 40th, and 45th partial sums of the power series

21

1( 3) .

2k

kk

xk

What is the radius of convergence?

Remember that each partial sum is a

polynomial. So we can plot it!

Some Questions Arise

Suppose that we consider the sorts of functions that are sums of Power Series:

•What are these functions like?

•Can we find formulas for them?

•Are power series functions continuous? Are they differentiable?

•If so, what do their derivatives and antiderivatives look like?

Consider an (already familiar) Example

You guessed it. . . It’s our old friend the geometric series!

2 3 4 5 6 71 x x x x x x x

We know it converges to whenever |x| < 1 and diverges elsewhere.

1

1 x

That is, 0

1( ) for all in 1,1 .

1n

n

f x x xx

Our first formula!

Near x = 1, the partial sums “blow up” giving us the asymptote we expect to see there.

Near x = -1the even and odd partial sums go opposite directions, preventing any convergence to the left of x = -1.

0

1( ) for all in 1,1 .

1n

n

f x x xx

We see the expected convergence on a “balanced” interval about x = 0.

This plot shows the 10th, 12th, 13th, and 15th partial sums of this series.

Clearly this function is both continuous and differentiable on its interval of convergence.

It is very tempting to say that the derivative for f (x) = 1 + x + x2 + x3 +. . .

should be

2 3 41 2 3 4 5x x x x

But is it? For that matter, does this series even converge? And if it does converge, what does it converge to?

Our first formula!

0

1( ) for all in 1,1 .

1n

n

f x x xx

What else can we observe?

It does converge, as the ratio test easily shows:

1( 2)

lim1

n

nn

n x

n x

So the ratio test limit is:

( 2)

lim | | 11n

nx x

n

So the “derivative” series also converges on (-1,1).

What about the endpoints? Does it converge at x = 1? x = -1?

The general form of the series is

2 3

0

1 2 3 4 1 n

n

x x x n x

Differentiating Power Series

6th partial sum 10th partial sum

2

1 1?

1 1

d

dx x x

The green graph is the partial sum, the red graph is 2

1

1 x

Does it converge to

This is true in general:

Let

be a power series with radius of convergence r > 0. And let

And

then

2 3 40 1 0 2 0 3 0 4 0( ) ( ) ( ) ( ) ( )S x a a x x a x x a x x a x x

2 31 2 0 3 0 4 0( ) 2 ( ) 3 ( ) 4 ( )D x a a x x a x x a x x

2 3 41 2 30 0 0 0 0( ) ( ) ( ) ( ) ( )

2 3 4

a a aA x a x x x x x x x x

•Both D and A converge with radius of convergence r.•On the interval (x0 - r,x0+r) S’(x) = D(x).•On the interval (x0 - r,x0+r) A’(x) = S(x).

Theorem: (Derivatives and Antiderivatives of Power Series)

Is this true of all Series Functions?

No. Power Series are very special in this regard!

If, instead of adding up powers of x with coefficients, we add up trig functions, we get very different behavior.

2 2 2 2

8 1 1 1sin( ) sin(3 ) sin(5 ) sin(7 )

3 5 7x x x x

Consider, for example

The partial sums of the series

look like this.

2 2 2 2

8 1 1 1sin( ) sin(3 ) sin(5 ) sin(7 )

3 5 7x x x x

1st partial sum 2nd partial sum 8th partial sum

3rd partial sum 10th partial sum5th partial sum

-2

-1

1

2

4 1 1 1sin( ) sin(3 ) sin(5 ) sin(7 )

3 5 7x x x x

The partial sums of the series

look like this.