1. Chapter 10 Power Series

2. 10.1 Approximating Functions with Polynomials

3. Slide 10 - 3 Linear approximation near x = a (1 st order) 2 nd order approximation near x = a

4. Slide 10 - 4 Example: Approximation of sin(x) near x = a (1 st order) (3 rd order) (5 th order)

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6. Slide 10 - 6 Brook Taylor 1685 - 1731 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

7. Slide 10 - 7 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 example:

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10. Slide 10 - 10 If we plot both functions, we see that near zero the functions match very well!

11. Slide 10 - 11 This pattern occurs no matter what the original function was! Our polynomial: has the form: or:

12. Slide 10 - 12 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:

13. Slide 10 - 13 Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:

14. Slide 10 - 14 The more terms we add, the better our approximation.

15. Slide 10 - 15 To find Factorial using the TI-83:

16. Slide 10 - 16 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:

17. Slide 10 - 17 Exercise 3: find the Taylor series for:

18. Slide 10 - 18 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.

19. Slide 10 - 19 3) Use the fourth degree Taylor polynomial of cos(2x) to find the exact value of Practice. 1) Show that the Taylor series expansion of e x is: 2) Use the previous result to find the exact value of:

20. 10.2 and 10.3 Properties of Power Series: Convergence

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23. Slide 10 - 23 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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25. Slide 10 - 25 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).

26. Slide 10 - 26 Common Taylor Series: