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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 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 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 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 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 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 Series • The first few partial sums are Graph of the Bessel function:
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 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 • 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
If we plot both functions, we see that near zero the functions match very well!
This pattern occurs no matter what the original function was! Our polynomial: has the form: or:
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:
Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
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:
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.
. 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
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].
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).
