Section 9.4b Radius of convergence

Definition: Absolute Convergence Recall the Direct Comparison Test from last class… To apply this test, the terms of the unknown series must be nonnegative. This doesn’t limit the usefulness of this test, because we can apply it to the absolute value of the series. Definition: Absolute Convergence If the series of absolute values converges, then converges absolutely. Thm: Absolute Convergence Implies Convergence If converges, then converges.

has no negative terms, and it is term-by-term less than or equal to: A quick example with this new rule… Show that the given series converges for all x. This series has no negative terms, and it is term-by-term less than or equal to: Which we know converges to e. Therefore, converges by direct comparison. Since converges absolutely, it converges.

Also recall this theorem from last class… There are three possibilities for with respect to convergence: 1. There is a positive number R such that the series diverges for but converges for . The series may or may not converge at either of the endpoints and . 2. The series converges for every x . 3. The series converges at x = a and diverges elsewhere The number R is the radius of convergence

Using the Ratio Test to find radius of convergence Find the radius of convergence of the given series. Check for absolute convergence using the Ratio Test:

Using the Ratio Test to find radius of convergence Find the radius of convergence of the given series. So the ratio of each term to the previous term: Which we need to be less than one: The series converges absolutely (and hence converges) on this interval, and diverges when x < –10 and for x > 10. The radius of convergence is 10

Practice Problems The radius of convergence is 1 Find the radius of convergence of the given power series. This is a geometric series with So it will only converge when The radius of convergence is 1

Practice Problems The radius of convergence is 1/3 Find the radius of convergence of the given power series. Ratio Test for absolute convergence: The series converges for or The radius of convergence is 1/3

Practice Problems 8 The radius of convergence is Find the radius of convergence of the given power series. Ratio Test for absolute convergence: The series converges for all values of x. 8 The radius of convergence is

Practice Problems Find the radius of convergence of the given power series. Ratio Test for absolute convergence:

Practice Problems The radius of convergence is 4 Find the radius of convergence of the given power series. Ratio Test for absolute convergence: The series converges for or The radius of convergence is 4

Practice Problems The radius of convergence is 0 Find the radius of convergence of the given power series. Ratio Test for absolute convergence: The series only converges for x = 4. The radius of convergence is 0

Practice Problems Interval of convergence: Find the interval of convergence of the given series and, within this interval, the sum of the series as a function of x. This is a geometric series with: It converges when: Interval of convergence:

Practice Problems Find the interval of convergence of the given series and, within this interval, the sum of the series as a function of x. This is a geometric series with: Sum

Practice Problems Interval of convergence: Find the interval of convergence of the given series and, within this interval, the sum of the series as a function of x. This is a geometric series with: It converges when: Which is always!!! Interval of convergence: Sum