1. TAYLOR AND MACLAURIN  how to represent certain types of functions as sums of power series  You might wonder why we would ever want to express a known function as a sum of infinitely many terms.  Integration. (Easy to integrate polynomials)  Finding limit  Finding a sum of a series (not only geometric, telescoping)

2. TAYLOR AND MACLAURIN Example: Maclaurin series ( center is 0 ) Example:Find Maclaurin series

3. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

4. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Example: Find Maclaurin series

5. TAYLOR AND MACLAURIN TERM-081

6. TAYLOR AND MACLAURIN TERM-091

7. TAYLOR AND MACLAURIN TERM-101

8. : TAYLOR AND MACLAURIN TERM-082

9. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN TERM-102

10. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN TERM-091

11. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Example: Find the sum of the series

12. TAYLOR AND MACLAURIN TERM-102

13. TAYLOR AND MACLAURIN TERM-082

14. TAYLOR AND MACLAURIN Leibniz’s formula: Example:Find the sum

15. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

16. The Binomial Series Example: DEF: Example:

17. The Binomial Series binomial series. NOTE:

18. The Binomial Series TERM-101 binomial series.

19. The Binomial Series TERM-092 binomial series.

20. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence Example: Find Maclaurin series

21. TAYLOR AND MACLAURIN TERM-102

22. TAYLOR AND MACLAURIN TERM-111

23. TAYLOR AND MACLAURIN TERM-101

24. TAYLOR AND MACLAURIN TERM-082

25. TAYLOR AND MACLAURIN Important Maclaurin Series and Their Radii of Convergence MEMORIZE: these Maclaurin Series

26. TAYLOR AND MACLAURIN Maclaurin series ( center is 0 ) Taylor series ( center is a )

27. TAYLOR AND MACLAURIN TERM-091

28. TAYLOR AND MACLAURIN TERM-092

29. TAYLOR AND MACLAURIN TERM-082

30. TAYLOR AND MACLAURIN Taylor series ( center is a ) Taylor polynomial of order n DEF:

31. TAYLOR AND MACLAURIN TERM-102 The Taylor polynomial of order 3 generated by the function f(x)=ln(3+x) at a=1 is: Taylor polynomial of order n DEF:

32. TAYLOR AND MACLAURIN TERM-101

33. TAYLOR AND MACLAURIN TERM-081

34. TAYLOR AND MACLAURIN Taylor series ( center is a ) Taylor polynomial of order n Remainder Taylor Series Remainder consist of infinite terms for some c between a and x. Taylor’s Formula REMARK: Observe that :

35. TAYLOR AND MACLAURIN for some c between a and x. Taylor’s Formula for some c between 0 and x. Taylor’s Formula

36. TAYLOR AND MACLAURIN Taylor series ( center is a ) nth-degree Taylor polynomial of f at a. DEF: Remainder DEF: Example:

37. TAYLOR AND MACLAURIN TERM-092

38. TAYLOR AND MACLAURIN TERM-081