11.8 Power Series

11.9Representations of Functions as

Power Series

11.10 Taylor and Maclaurin Series

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Power Series

A power series is a series of the form

where x is a variable and the cn’s are constants called the coefficients of the series.

A power series may converge for some values of x and diverge for other values of x.

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Power Series

The sum of the series is a function:

f (x) = c0 + c1x + c2x2 + . . . + cnxn + . . .

whose domain is the set of all x for which the series converges. Notice that f resembles a polynomial. The only difference is that f has infinitely many terms.

Note: if we take cn = 1 for all n, the power series becomes the geometric series

xn = 1 + x + x2 + . . . + xn + . . .

which converges when –1 < x < 1 and diverges when | x | 1.

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Power Series

More generally, a series of the form

is called a power series in (x – a) or a power series centered at a or a power series about a.

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Power Series

The main use of a power series is to provide a way to represent some of the most important functions that arise in mathematics, physics, and chemistry.

Example: the sum of the power series,

, is called a Bessel function.

•Electromagnetic waves in a cylindrical waveguide•Pressure amplitudes of inviscid rotational flows•Heat conduction in a cylindrical object•Modes of vibration of a thin circular (or annular) artificial membrane•Diffusion problems on a lattice•Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle•Solving for patterns of acoustical radiation•Frequency-dependent friction in circular pipelines•Signal processing

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Power Series

The first few partial sums are

Graph of the Bessel function:

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Power Series: convergence

The number R in case (iii) is called the radius of convergence of the power series.

This means: the radius of convergence is R = 0 in case (i) and R = in case (ii).

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Power Series

The interval of convergence of a power series is the interval of all values of x for which the series converges.

In case (i) the interval consists of just a single point a.

In case (ii) the interval is ( , ).

In case (iii) note that the inequality | x – a | < R can be rewritten as

a – R < x < a + R.

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Representations of Functions as Power Series

Example: Consider the series

We have obtained this equation by observing that the series is a geometric series with a = 1 and r = x.

We now regard Equation 1 as expressing the function

f (x) = 1/(1 – x) as a sum of a power series.

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Approximating Functions with Polynomials

1111

Example: Approximation of sin(x) near x = a

(1st order)(3rd order)

(5th order)

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Brook Taylor1685 - 1731

Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.

Greg Kelly, Hanford High School, Richland, Washington

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Suppose we wanted to find a fourth degree polynomial of the form:

2 3 40 1 2 3 4P x a a x a x a x a x

ln 1f x x at 0x that approximates the behavior of

If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.

0 0P f

Practice:

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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x

ln 1f x x

0 ln 1 0f

2 3 40 1 2 3 4P x a a x a x a x a x

00P a 0 0a

1

1f x

x

10 1

1f

2 31 2 3 42 3 4P x a a x a x a x

10P a 1 1a

2

1

1f x

x

10 1

1f

22 3 42 6 12P x a a x a x

20 2P a 2

1

2a

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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x

3

12

1f x

x

0 2f

3 46 24P x a a x

30 6P a 3

2

6a

4

4

16

1f x

x

4 0 6f

4424P x a

440 24P a 4

6

24a

2

1

1f x

x

10 1

1f

22 3 42 6 12P x a a x a x

20 2P a 2

1

2a

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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x

2 3 41 2 60 1

2 6 24P x x x x x

2 3 4

02 3 4

x x xP x x ln 1f x x

P x

f x

If we plot both functions, we see that near zero the functions match very well!

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This pattern occurs no matter what the original function was!

Our polynomial: 2 3 41 2 60 1

2 6 24x x x x

has the form: 42 3 40 0 0

0 02 6 24

f f ff f x x x x

or: 42 3 40 0 0 0 0

0! 1! 2! 3! 4!

f f f f fx x x x

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Maclaurin Series:

(generated by f at )0x

2 30 00 0

2! 3!

f fP x f f x x x

If we want to center the series (and it’s graph) at zero, we get the Maclaurin Series:

Taylor Series:

(generated by f at )x a

2 3

2! 3!

f a f aP x f a f a x a x a x a

Definition:

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2 3 4 5 61 0 1 0 1

1 0 2! 3! 4! 5! 6!

x x x x xP x x

cosy x

cosf x x 0 1f

sinf x x 0 0f

cosf x x 0 1f

sinf x x 0 0f

4 cosf x x 4 0 1f

2 4 6 8 10

1 2! 4! 6! 8! 10!

x x x x xP x

Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:

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cosy x 2 4 6 8 10

1 2! 4! 6! 8! 10!

x x x x xP x

The more terms we add, the better our approximation.

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To find Factorial using the TI-83:

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cos 2y xRather than start from scratch, we can use the function that we already know:

2 4 6 8 102 2 2 2 2

1 2! 4! 6! 8! 10!

x x x x xP x

Exercise 2: find the Taylor polynomial approximation at 0 (Maclaurin series) for:

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cos at 2

y x x

2 3

0 10 1

2 2! 2 3! 2P x x x x

cosf x x 02

f

sinf x x 12

f

cosf x x 02

f

sinf x x 12

f

4 cosf x x 4 02

f

3 5

2 2

2 3! 5!

x xP x x

Exercise 3: find the Taylor series for:

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When referring to Taylor polynomials, we can talk about number of terms, order or degree.

2 4

cos 12! 4!

x xx This is a polynomial in 3 terms.

It is a 4th order Taylor polynomial, because it was found using the 4th derivative.

It is also a 4th degree polynomial, because x is raised to the 4th power.

The 3rd order polynomial for is , but it is degree 2.cos x2

12!

x

The x3 term drops out when using the third derivative.

This is also the 2nd order polynomial.

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3) Use the fourth degree Taylor polynomial of   cos(2x)    to find the exact value of

Practice example:

                                                  .

1) Show that the Taylor series expansion of ex is:

2) Use the previous result to find the exact value of:

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Common Taylor Series:

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Properties of Power Series:Convergence

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3131

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Convergence of Power Series:

0

)(n

nn axc

1

lim

n

n

n c

cR

The center of the series is x = a. The series converges on the open interval and may converge at the endpoints. ),( RaRa

The Radius of Convergence for a power series is:

is

You must test each series that results at the endpoints of the interval separately for convergence.

Examples: The series is convergent on [-3,-1]

but the series is convergent on (-2,8].

02)1(

)2(

n

n

n

x

0 15

)3()1(

nn

nn

n

x

3333

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Convergence of Taylor Series: is

If f has a power series expansion centered at x = a, then the

power series is given by

And the series converges if and only if the Remainder satisfies:

0

)(

)(!

)()(

n

nn

axn

afxf

Where: is the remainder at x, (with c between x and a).

0)(lim

xRnn

1)!1(

)( )()()1(

n

ncf

n axxRn