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Section 8.8: Taylor Series

Section 8.8: Taylor Series. (So cool it needs its own theme music). Recall . This is valid on the interval of convergence of the original series. So what is ?? . It must be e x !!. Theorem. If f(x) can be written as a power series it must be.

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Section 8.8: Taylor Series

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  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 y = x

  20. Sin(x) and its MacLaurin Series

  21. Sin(x) and its MacLaurin Series

  22. Sin(x) and its MacLaurin Series

  23. Cos(x) and its MacLaurin Series y = 1

  24. Cos(x) and its MacLaurin Series

  25. Cos(x) and its MacLaurin Series

  26. Cos(x) and its MacLaurin Series

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