9.10 More Power Series

Example 2

Example 2

Problem 6

Find the a power series for the function centered at c.

____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

Problem 10

Find the a power series for the function centered at c.

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

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 xn (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

The first 3 properties restated

The Series for

Sinx = x - x3 + x5 - x7… 3! 5! 7!

If we replace x by x-3 on both side will we still get a true equation?

The Series for

Sinx = x - x3 + x5 - x7… 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.

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

Recall

AP does not usually ask these questions:

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