1 / 20

Power Series is an infinite polynomial in x

Power Series is an infinite polynomial in x. Is a power series centered at x = 0. and. Is a power series centered at x = a. Examples of Power Series . Is a power series centered at x = 0. and. Is a power series centered at x = -1. Geometric Power Series.

malcolm
Download Presentation

Power Series is an infinite polynomial in x

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Power Series is an infinite polynomial in x Is a power series centered at x = 0. and Is a power series centered at x = a.

  2. Examples of Power Series Is a power series centered at x = 0. and Is a power series centered at x = -1.

  3. Geometric Power Series

  4. The graph of f(x) = 1/(1-x) and four of its polynomial approximations .

  5. Convergence of a Power Series There are three possibilities 1)There is a positive number R such that the series diverges for |x-a|> R but converges for |x-a|< R. The series may or may not converge at the endpoints, x = a - R and x = a + R. 2)The series converges for every x. (R = .) 3)The series converges at x = a and diverges elsewhere. (R = 0)

  6. Review of tests for convergence 2. Using the ratio test: a) Series converges for L < 1 b) Series diverges for L >1 c) Test is inconclusive if L = 1 3. Using the root test: a) Series converges for L < 1 b) Series diverges for L >1 c) Test is inconclusive if L = 1 • Geometric Series • converges for | r | <1 • diverges for | r | >1 and | r | = 1

  7. What is the interval of convergence? Since r = x, the series converges |x| <1, or -1 < x < 1. Test endpoints of –1 and 1. Series diverges Series diverges 1 -1 interval of convergence is (-1,1).

  8. Geometric Power Series 1.Find the function 2. Find the radius of convergence R = 3

  9. 3. Find the interval of convergence For x = -2, Test endpoints Geometric series with r < 1, converges For x = 4 By nth term test, the series diverges. Interval of convergence

  10. Find interval of convergence Use the ratio test: 0 < 1 for all reals R=  Series converges for all reals. (-, ) Interval of convergence

  11. Finding interval of convergence Use the ratio test: R=0 Series converges only for center point [0, 0] Interval of convergence

  12. Finding interval of convergence Use the ratio test: (-1, 1) -1< x <1 R=1 Test endpoints For x = -1 For x = 1 Harmonic series diverges Alternating Harmonic series converges [-1, 1) Interval of convergence

  13. Differentiation and Integration of Power Series If the function is given by the series Has a radius of convergence R > 0, on the interval (c-R, c+R) the function is continuous, differentiable and integrable where: and The radius of convergence is the same but the interval of convergence may differ at the endpoints.

  14. Constructing Power Series If a power series exists has a radius of convergence = R It can be differentiated So the nth derivative is

  15. Finding the coefficients for a Power Series All derivatives for f(x) must equal the series Derivatives at x = a.

  16. If f has a series representation centered at x=a, the series must be If f has a series representation centered at x=0, the series must be

  17. Form a Taylor Polynomial of order 3 for sin x at a =

  18. The graph of f(x) = ex and its Taylor polynomials

  19. Find the derivative and the integral

  20. Taylor polynomials for f(x) = cos (x)

More Related