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

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

33 Power Series The sum of the series is a function: f (x) = c 0 + c 1 x + c 2 x c n x n +... 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 c n = 1 for all n, the power series becomes the geometric series x n = 1 + x + x x n +... which converges when –1 < x < 1 and diverges when | x |  1.

44 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.

55 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 particleSchrödinger equation Solving for patterns of acoustical radiation Frequency-dependent friction in circular pipelines Signal processing

66 Power Series The first few partial sums are Graph of the Bessel function:

77 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).

88 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.

99 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.

10 Approximating Functions with Polynomials

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

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13 Brook Taylor 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

14 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. Practice:

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

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

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

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

21 The more terms we add, the better our approximation.

22 To find Factorial using the TI-83:

23 Rather than start from scratch, we can use the function that we already know: Exercise 2: find the Taylor polynomial approximation at 0 (Maclaurin series) for:

24 Exercise 3: find the Taylor series for:

25 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 x 3 term drops out when using the third derivative. This is also the 2nd order polynomial.

26 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 e x is: 2) Use the previous result to find the exact value of:

27 Common Taylor Series:

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29 Properties of Power Series: Convergence

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32 Convergence of Power Series: The center of the series is x = a. The series converges on the open interval and may converge at the endpoints. 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].

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34 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).