1. Taylor Series

2. Theorem

3. Definition The series is called the Taylor series of f about c (centered at c)

4. Definition The series is called the Maclaurin series of f about c (centered at c) Thus a Maclaurin series is a Taylor series centered at 0

5. Examples I

6. Example (1)Taylor Series for f(x) = sinx about x = 2π

7. Taylor Series for sinx about 2π

8. Example (2)Taylor Series for f(x) = sinx about x = π

9. Taylor Series for sinx about π

10. Example (3)Taylor Series for f(x) = sinx about x = π/2

11. Taylor Series for sinx about π/2

12. Example (4)Maclaurin Series for f(x) = sinx

13. Maclaurin Series for sinx

14. Example(5)The Maclaurin Series for f(x) = x sinx

15. Examples II

16. Example (1)Maclaurin Series for f(x) = ex

17. Maclaurin Series for ex

18. Example (2) Find a power series for the function g(x) =

20. Example (3)TaylorSeries for f(x) = lnx about x=1

21. Taylor Series for lnx about x = 1

22. I-1Taylor Series for f(x) = cos2x about x = 5π

23. Taylor Series for cos2x about 5π

24. I-2Taylor Series for f(x) = cosx about x = 3π/4

25. Taylor Series for cosx about 3π/4

26. I.3Taylor Series for f(x) = cosx about x = π/6

27. Taylor Series for cosx about π/6

28. I.4Taylor Series for f(x) = e3x about x = 5

30. Homework