Representation of Functions by Power Series

1. Representation of Functions by Power Series Lesson 9.9

2. Geometric Power Series • Consider the function • Note the similarity to the sum of the geometric series

3. Geometric Power Series • Thus we could let a = 1 and r = x and have • Where is this power series centered? • At x = 0, for the interval -1 < x < 1 • Note also that our original f(x) is defined for all x ≠ 1

4. Geometric Power Series • To center this series at x = -3 you could specify • Then • a = ¼ • r = [(x + 3)/4] • So convergence would be for the interval

5. Geometric Power Series • Try this out … find a power series for the function centered at c

6. Operations with Power Series • Note the following rules

7. Adding Two Power Series • Consider a function which can be broken into two using partial fractions • Now determine two geometric power series which are added to give us a series for the original function

8. Try Another • Try these … just for practice See Example 4 and Example 2

9. Assignment • Lesson 9.9 • Page 674 • Exercises 5 – 25 odd