Section 8.8: Taylor Series

# Section 8.8: Taylor Series

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## Section 8.8: Taylor Series

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##### Presentation Transcript

1. Section 8.8: Taylor Series (So cool it needs its own theme music)

2. Recall This is valid on the interval of convergence of the original series.

3. So what is ?? It must be ex!!

4. Theorem If f(x) can be written as a power series it must be This is the Taylor series for f. When c = 0,it is called the Maclaurin series for f.

5. Why? Suppose What is f(c) ??

6. What is f’(c) ??

7. What is f’’(c) ??

8. What is f’’’(c) ??

9. Find its Maclaurin Series.

10. Find its Maclaurin Series.

11. Find its Maclaurin Series.

12. Summary So Far

13. Converges by ratio test What does it converge to? e2

14. Mystery Solved: Why does L’Hopital’s Rule work? Why is ?

15. Geometric series

16. Notice that

17. Euler’s Formula Corollary:

18. What Taylor Series Really Do

19. Sin(x) and its MacLaurin Series

20. Sin(x) and its MacLaurin Series

21. Sin(x) and its MacLaurin Series

22. Cos(x) and its MacLaurin Series

23. Cos(x) and its MacLaurin Series

24. Cos(x) and its MacLaurin Series