1. Lecture 25 – Power Series Def: The power series centered at x = a:
x is the variable and the c’s are constants (coefficients)

2. For any power series, exactly one of the following is true:

3. Example 1 – Radius and Interval of Convergence
Ratio Test:
Series converges for

4. Example 2 – Radius and Interval of Convergence
Ratio Test:
Series converges for

5. Example 3 – Radius and Interval of Convergence
Ratio Test:

6. Example 3 – continued (testing endpoints)

7. Example 4 – Radius and Interval of Convergence
Root Test:

8. Example 4 – continued (testing endpoints)

9. Example 5 – Radius and Interval of Convergence
Geometric Series:

10. Example 5 – continued – what is the converging value?
Geometric Series:

11. Lecture 26 – More Power Series
The geometric series:
As a power series with a = 1, r = x and cn = 1 for all n:
In other words, the function f(x) can be written as a power series.
with

12. Example 1 Create new power series for other functions through: and
sum, difference, multiplication, division, composition
and
differentiation and integration
Example 1
with

13. Example 2
with

14. Consider the graphs:

15. Example 3
with

16. Need to solve for C. Set x = 0 to get:
Test endpoints???

17. Example 4
with

18. Need to solve for C. Set x = 0 to get:
Test endpoints???

19. Lecture 27 – Approximating Functions
Consider the following:
Now, use the reciprocal function and tangent line to get an approximation. 1 2 3

20. 2
2.01

21. First derivative gave us more information about the function (in particular, the direction).
For values of x near a the linear approximation given by the tangent line should be better than the constant approximation.
Second derivative will give us more information (curvature).
For values of x near a the quadratic approximation should be better than the linear approximation.

22. What quadratic is used as the approximation?
Key idea: Need to have quadratic match up with the function and its first and second derivatives at x = a.

23. Use p2(x) to get a better approximation.
2.01

24. Graphical Example at x = 01 2 3

25. What higher degree polynomial is appropriate?
Key idea: Need to have nth degree polynomial match up with the function and all of its derivatives at x = a.

26. The coefficients, ck, for the nth degree Taylor polynomial approximating the function f(x) at x = a have the form:

27. Lecture 28 – Taylor Polynomials
Def: The Taylor polynomial of order n for function f at x = a:
The remainder term for using this polynomial:
for some c between x and a.
where M provides a bound on
how big the n+1st derivative
could possibly be.

28. Estimate the maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3].

30. What is the actual maximum error in approximating the reciprocal function at x = 2 with an 8th order Taylor polynomial on the interval [2, 3]?

31. What nth degree polynomial would you need in order to keep the error below .0001?

32. To keep error below .0001, need to keep Rn below .0001.

33. Lecture 29 – Taylor Series
The Taylor series centered at x = a:
is a power series with
The Taylor series centered at x = 0 is called a Maclaurin series:

34. Example 1
Find the Maclaurin series for f (x) = sin x.

35. Example 2
Find the Maclaurin series for f (x) = ex.

36. For what values of x will the last two series converge?
Ratio Test:
Series converges for
Series converges for

37. Consider the graphs:

38. Example 3
Find the Maclaurin series for f (x) = ln(1 + x).

39. For what values of x will the series converge?

40. Example 4
Creating new series for:

41. Lecture 30 – More Taylor Series
Create and use other Taylor series like was done with power series.

42. Example 1

43. Example 2

44. Example 3

45. Example 4

46. Example 5