Section 6: Power Series & the Ratio Test

Power Series: centered at x = a cn’s are just coefficients which depend on n

Interval of Convergence: the set of x-values for which the series converges Radius of Convergence(R): the distance from the center of the interval to either endpoint

3 Cases: R = 0 (series converges only at the center) R =  (series converges everywhere) R = some positive number |x – a| < R (series converges when a – R < x < a + R )

Ex 1: The power series converges at x = 6 and diverges at x = 8 Ex 1: The power series converges at x = 6 and diverges at x = 8. State whether the series converges, diverges, or cannot be determined at the x-values: -8, -6, -2, -1, 0, 2, 5, 7, & 9

Ex 2: Find the radius of convergence of the two power series: a) b)

The Ratio Test: Let an be a series in which an > 0 for all n.

Ex 3: Use the ratio test to determine convergence: a) b) c)

Ex 4: Determine the radius of convergence of the power series:

Section 6 WS #1-17 odds, 18,20, 21-25 OODS, 27-39 eoo