11.8 11.9 11.10 Taylor and Maclaurin Series Representations of Functions as Power Series Taylor and Maclaurin Series 11.10

Reminder: 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.

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. Recall the example: 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.

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].

Radius and Interval of Convergence: Examples:

Why do we use 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. Many applications: EM waves in cylindrical waveguide, heat conduction Electronic and signal processing Modes of vibration of artificial membranes, Acoustics. https://www.youtube.com/watch?v=ewIr4lO4408

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.

Example Express 1/(1 + x2) as the sum of a power series and find the interval of convergence. Solution: Replacing x by –x2 in Equation 1, we have Because this is a geometric series, it converges when | –x2 | < 1, that is, x2 < 1, or | x | < 1.

Example – Solution Therefore the interval of convergence is (–1, 1). cont’d Therefore the interval of convergence is (–1, 1). (Of course, we could have determined the radius of convergence by applying the Ratio Test, but that much work is unnecessary here.)

Differentiation and Integration of Power Series

We use the previous facts to actually find the expansion of a function as a power series….

Brook Taylor 1685 - 1731 “Taylor” Series 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

Taylor Polynomial expansion of a function

Example: Approximation of sin(x) near x = 0 (3rd order) (1st order) (5th order)

How does it work…? Suppose we wanted to find a fourth degree polynomial of the form: at 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.

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

Our polynomial: has the form: This pattern occurs no matter what the original function was! or:

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

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

The more terms we add, the better our approximation.

To find Factorial using the TI-83:

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:

Exercise 3: find the Taylor series for:

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 3rd order polynomial for is , but it is degree 2. The x3 term drops out when using the third derivative. This is also the 2nd order polynomial.

Practice: work these now                                                   . Practice: work these now 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

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).

Common Taylor Series: