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# 11.8 - PowerPoint PPT Presentation

11.8. Power Series. 11.9. Representations of Functions as Power Series. Taylor and Maclaurin Series. 11.10. Power Series. A power series is a series of the form where x is a variable and the c n ’s are constants called the coefficients of the series.

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Power Series

11.9

Representations of Functions as Power Series

Taylor and Maclaurin Series

11.10

• 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.

• 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.

• 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.

• We will see that the main use of a power series is that it provides 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

• The first few partial sums are

Graph of the Bessel function:

• 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).

• The interval of convergence of a power series is the interval that consists 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.

• We start with an equation that we have seen before:

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

• But here our point of view is different. We now regard Equation 1 as expressing the function f(x) = 1/(1 – x) as a sum of a power series.

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

(3rd order)

(1st order)

(5th order)

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.

Brook Taylor

1685 - 1731

Greg Kelly, Hanford High School, Richland, Washington

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

Practice:

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

at

that approximates the behavior of

Our polynomial:

has the form:

or:

Maclaurin was! Series:

Taylor Series:

(generated by f at )

(generated by f at )

Definition:

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

To find was!Factorial using the TI-83:

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

Rather than start from scratch, we can use the function that we already know:

Exercise 3: was!find the Taylor series for:

The was!3rd order polynomial for is , but it is degree 2.

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

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 x3 term drops out when using the third derivative.

This is also the 2nd order polynomial.

. was!

Practice example:

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

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

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

### Properties of Power Series: was!Convergence

is

The Radius of Convergence for a power series is:

• The center of the series is x = a. The series converges on the open interval and may converge at the endpoints.

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].

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:

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

Common was!Taylor Series: