1. Review of Power Series and Taylor Polynomials

2. Infinite Sums and Power Series Recall Infinite Sums:

3. Infinite Sums and Power Series Recall Infinite Sums:

4. Infinite Sums and Power Series Recall Infinite Sums: In General:

5. Infinite Sums and Power Series In General: Three possible outcomes of infinite sums: or Diverges Converges Neither

6. Infinite Sums and Power Series In General: Special Type of Infinite Sum: Power Series Which gives

7. Infinite Sums and Power Series In General: Special Type of Infinite Sum: Power Series Depends on x

8. Infinite Sums and Power Series In General: Special Type of Infinite Sum: Power Series Depends on x

9. Infinite Sums and Power Series In General: Special Type of Infinite Sum: Power Series Depending on x, can either diverge, converge, or neither

10. Example First 10 Terms

11. Example First 20 Terms

12. Example First 50 Terms

13. Example First 100 Terms

14. Example Eventuall y

15. Determining Interval of Convergence Ratio Test Converges if Diverges if If, test is inconclusive.

16. Determining Interval of Convergence Now Note If: The n

17. Determining Interval of Convergence Ratio Test Converges if Diverges if If, test is inconclusive.

18. Determining Interval of Convergence Ratio Test Converges if Diverges if If, test is inconclusive.

19. Determining Interval of Convergence Ratio Test Converges if Diverges if If, test is inconclusive.

20. Determining Interval of Convergence Ratio Test Converges if Diverges if If, test is inconclusive. Is Known as “Radius of Convergence”

21. Determining Interval of Convergence Interval of Convergence Radius of Convergence

22. Power Series Manipulation Sum : Derivative: Reindexing:

23. Taylor Series For a Power Series It is always true that So given a function, we can write it as a power series. A power series that describes a function is called a “Taylor Series”

24. Taylor Series For a Power Series For Example

25. Taylor Series

29. Taylor Polynomials If we kept going, we would find no error! KEY IDEA: We can use Taylor Series (or Power Series) as substitutes for common functions, because they ARE THE SAME THING

30. Questions?