1. 9.10 More Power Series

2. Example 2

4. Problem 6 Find the a power series for the function centered at c.

5. ____4____ 5+3 – (x+3) Geometric series with ___4___ first term ½ and r = (x+3)/8 8 – (x+3) __1/2___ 1 – (x+3) ½ + 1/2 (x+3)/8 + 1/2 ((x+3)/8) 2 +… 8 We want the form

6. Problem 10 Find the a power series for the function centered at c.

7. Problem 10 solution ___-3____Geometric series with 1 – 2x a= 1 r= -2/3(x -2) ___-3____ 1- 4 – 2(x -2) ___-3____ -3 –2(x -2) ___ 1____ 1 - 2/3(x -2) We want the form

8. If we have a known power series what operations can I use to create a new series? You may do any of the following 1. replace x by kx (on both sides of the equation) 2. replace x by x n (on both sides of the equation) 3. Add two known series 4. Take the derivative of a known series 5. Find a definite integral from 0 to x of a known series

9. The first 3 properties restated

10. The Series for Sinx = x - x 3 + x 5 - x 7 … 3! 5! 7! If we replace x by x-3 on both side will we still get a true equation?

11. The Series for Sinx = x - x 3 + x 5 - x 7 … 3! 5! 7! If we replace x by x-3 on both side will we still get a true equation? No, the left side would equal to sinx shifted 3 the right but the left side would not.

12. Note: to create other series with non zero centers such as y= sinx centered at π/2 Use the Taylor series to create the series

13. Recall

20. AP does not usually ask these questions:

23. p. 687 1-5 odd, 27-39 odd p. 676 5,7,12