1. Section 8.7: Power Series

2. If c = 0, Definition is a power series centered at c IMPORTANT: A power series is a function. Its value and whether or not it converges depends on which x you plug in. Think of them as “Long Polynomials”

3. If you plug in a number for x, you get a geometric series. It converges if |x|<1. Remember, a power series is a function.

4. By ratio test the series converges absolutely when Find all values of x at which the power series converges.

5. If x-5 is positive: If x-5 is negative:

6. |x-5| is really just the distance from x to 5 Let’s try geometry instead of algebra. 53 2 2 7 The Radius of Convergence is 2.

7. By ratio test converges absolutely when 3 < x < 7. But we must still check endpoints because the ratio test tells us nothing at x = 3 and x = 7. Diverges Radius of convergence R = 2 Interval of convergence [3, 7) Converges

8. Try the Root Test this time. So the series converges only at x = 2 Radius of convergence is 0 Unless x = 2

9. So the series converges no matter what x is. Radius of convergence = Interval of convergence=

10. Theorem For a power series exactly one of these possibilities occurs: 1.It converges only at x = c 2.It converges at every number x 3.There is a number R > 0 so that a)It converges absolutely if |x-c|<R b)It diverges if |x-c|>R

11. Suppose converges at x = 8 Does it converge at x = 6 ? Does it converge at x = -2 ? Does it converge at x = -1 ? 3-28 Does it converge at x = 9? Yes ! Maybe ?

12. The Calculus of Power Series If a power series is a function, can we integrate or differentiate it? How much like a polynomial is it?

13. Theorem This is valid on the interval of convergence of the original series.

14. So what is ?? It must be e x !!