1. Alternating Series An alternating series is a series where terms alternate in sign.

2. Thm. Alternating Series Test The alternating series Ʃ (-1) n a n will converge if i) ii) a n is a decreasing sequence (a n + 1 < a n )

3. Ex.

5. Thm. Absolute Convergence Test If Ʃ |a n | converges, then Ʃ a n converges. Ex.

6. Ʃ a n is absolutely convergent if Ʃ |a n | and Ʃ a n both converge. Ʃ a n is conditionally convergent if Ʃ a n converges but Ʃ |a n | diverges.

7. Ex.

9. Thm. Alternating Series Remainder If you use N terms to approximate the sum of the convergent alternating series Σ(-1) n a n, then the error is less than a N + 1.

10. Ex. Approximate the value of using the first 6 terms. How many terms are needed to ensure an error less than.001?

11. Pract. Determine the convergence, and state the test used 1. 2. 3. Find with an error less than.001 Conv., Alt. Series Test Div., n-th term test 0.368