Clicker Question 1 The series A. converges to a number greater than 1/5. B. converges to a number less than 1/5. C. diverges.

Clicker Question 2 The series A. converges to a number greater than 1/5. B. converges to a number less than 1/5. C. diverges.

Absolute Convergence and the Ratio Test (11/22/13) A series an is said to converge absolutely if the series |an| converges. For example, the alternating harmonic series (-1)n+1(1/n) converges but does not converge absolutely. Obviously, if a series consists of positive terms to start with, convergence and absolute convergence are the same thing. If a series converges but does not converge absolutely, it is said to converge conditionally.

Clicker Question 3 The series A. converges absolutely. B. converges but does not converge absolutely. C. converges absolutely but does not converge. D. diverges.

The Ratio Test We know that a geometric series converges if its ratio x satisfies that |x| < 1. If a series is not geometric, we can still compute the ratio of (the absolute values of) two adjacent terms and then ask if that (changing) ratio is approaching a limit as n goes to . If that limit exists, guess what values will indicate convergence?

The Ratio Test (continued) If the ratio of the absolute values of the individual terms is approaching a number L as n  , then: 1. If L < 1, the series converges absolutely. 2. If L > 1, the series diverges. 3. If L = 1, the test tells you nothing! For example, 1/2 + 2/4 + 3/8 + … + n/2n +… must converge. Why? (Check!) Does this test show that 1 + 1/4 + 1/9 + 1/16 + … converges?

Clicker Question 4 The series A. Converges absolutely B. Converges conditionally C. Diverges

Assignment for Monday Read Section 11.6 to the middle of page 736 and do Exercises 1-19 odd and 35.