9.1 Power Series

1. 9.1 Power Series Start with a square one unit by one unit: 1 This is an example of an infinite series. This series converges (approaches a limiting value.) 1 Many series do not converge:

2. 9.1 Power Series If Sn has a limit as , then the series converges, otherwise it diverges. In an infinite series: a1, a2,… are terms of the series. an is the nth term. Partial sums: nth partial sum

3. 9.1 Power Series This converges to if , and diverges if . is the interval of convergence. Geometric Series: In a geometric series, each term is found by multiplying the preceding term by the same number, r.

4. 9.1 Power Series a r

5. 9.1 Power Series a r

6. 9.1 Power Series A power series is in this form: or The coefficientsc0, c1, c2… are constants. The center “a” is also a constant. (The first series would be centered at the origin if you graphed it. The second series would be shifted left or right. “a” is the new center.)

7. 9.1 Power Series If then If and we let , then: The partial sum of a geometric series is:

8. 9.1 Power Series We could generate this same series for with polynomial long division:

9. 9.1 Power Series multiply both sides by x. To find a series for Once we have a series that we know, we can find a new series by doing the same thing to the left and right hand sides of the equation. This is a geometric series where r=-x.

10. 9.1 Power Series So: Given: find: We differentiated term by term.

11. 9.1 Power Series hmm? Given: find:

12. 9.1 Power Series

13. 9.1 Power Series The previous examples of infinite series approximated simple functions such as or . This series would allow us to calculate a transcendental function to as much accuracy as we like using only pencil and paper!

14. 9.2 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

15. 9.2 Taylor Series If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation. Suppose we wanted to find a fourth degree polynomial of the form: that approximates the behavior of at

16. 9.2 Taylor Series

17. 9.2 Taylor Series

18. 9.2 Taylor Series If we plot both functions, we see that near zero the functions match very well!

19. 9.2 Taylor Series This pattern occurs no matter what the original function was! Our polynomial: has the form: or:

20. 9.2 Taylor Series Maclaurin Series: (generated by f at )

21. 9.2 Taylor Series Taylor Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:

22. 9.2 Taylor Series

23. 9.2 Taylor Series The more terms we add, the better our approximation.

24. 9.2 Taylor Series Rather than start from scratch, we can use the function that we already know:

25. 9.2 Taylor Series example:

26. 9.2 Taylor Series There are some Maclaurin series that occur often enough that they should be memorized. They are on pg 477

27. 9.2 Taylor Series

28. 9.2 Taylor Series When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 terms.

29. 9.2 Taylor Series The 3rd order polynomial for is , but it is degree 2. 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.

30. 9.2 Taylor Series List the function and its derivatives.

31. 9.2 Taylor Series Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x.

32. 9.2 Taylor Series We could generate this same series for with polynomial long division:

33. 9.2 Taylor Series This is a geometric series with a = 1 and r = -x.

34. 9.2 Taylor Series We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly. They do help to explain where the formula for the sum of an infinite geometric comes from. We will find other uses for these series, as well. A more impressive use of Taylor series is to evaluate transcendental functions.

35. 9.2 Taylor Series Both sides are even functions. Cos (0) = 1 for both sides.

36. 9.2 Taylor Series Both sides are odd functions. Sin (0) = 0 for both sides.

37. 9.2 Taylor Series and substitute for , we get: If we start with this function: This is a geometric series with a = 1 and r = -x2.

38. 9.2 Taylor Series If we integrate both sides: This looks the same as the series for sin (x), but without the factorials.

39. 9.2 Taylor Series

40. 9.3 Taylor’s Theorem Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use the calculator or computer to calculate directly.) An estimate is only useful if we have an idea of how accurate the estimate is. When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is.

41. 9.3 Taylor’s Theorem Since the truncated part of the series is: , the truncation error is , which is . Use to approximate over . For a geometric series, this is easy: When you “truncate” a number, you drop off the end. Of course this is also trivial, because we have a formula that allows us to calculate the sum of a geometric series directly.

42. 9.3 Taylor’s Theorem Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I:

43. 9.3 Taylor’s Theorem Lagrange Form of the Remainder Remainder after partial sum Sn where c is between a and x.

44. 9.3 Taylor’s Theorem Note that this looks just like the next term in the series, but “a” has been replaced by the number “c” in . This seems kind of vague, since we don’t know the value of c, but we can sometimes find a maximum value for . This is also called the remainder of order n or the error term.

45. 9.3 Taylor’s Theorem If M is the maximum value of on the interval between a and x, then: Remainder Estimation Theorem Lagrange Form of the Remainder Note that this is not the formula that is in our book. It is from another textbook. We will call this the Remainder Estimation Theorem.

46. 9.3 Taylor’s Theorem Prove that , which is the Taylor series for sinx, converges for all real x.

47. 9.3 Taylor’s Theorem Remainder Estimation Theorem Since the maximum value of sin x or any of it’s derivatives is 1, for all real x, M = 1. so the series converges.

48. 9.3 Taylor’s Theorem On the interval , decreases, so its maximum value occurs at the left end-point. Find the Lagrange Error Bound when is used to approximate and . Remainder after 2nd order term

49. 9.3 Taylor’s Theorem Remainder Estimation Theorem On the interval , decreases, so its maximum value occurs at the left end-point. error Find the Lagrange Error Bound when is used to approximate and . Error is less than error bound. Lagrange Error Bound

50. 9.3 Taylor Series We have saved the best for last!