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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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11 8

11.8

Power Series

11.9

Representations of Functions as Power Series

Taylor and Maclaurin Series

11.10


Power series

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.


Power series1

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.


Power series2

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.


Power series3

Power Series

  • 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


Power series4

Power Series

  • The first few partial sums are

Graph of the Bessel function:


Power series convergence

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).


Power series5

Power Series

  • 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.


Representations of functions as power series

Representations of Functions as Power Series

  • 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.


Approximating functions with polynomials

Approximating Functions with Polynomials


11 8

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

(3rd order)

(1st order)

(5th order)


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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


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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


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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:

has the form:

or:


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Maclaurin 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:


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


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The more terms we add, the better our approximation.


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To find Factorial using the TI-83:


11 8

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

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


11 8

Exercise 3:find the Taylor series for:


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The 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.


11 8

.

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 convergence

Properties of Power Series:Convergence


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Convergence of Power Series:

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].


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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:

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


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Common Taylor Series:


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