1. Alternating Series Lesson 9.5

2. Alternating Series Two versions When odd-indexed terms are negative When even-indexed terms are negative

3. Alternating Series Test Recall does not guarantee convergence of the series In case of alternating series … Must converge if { a k } is a decreasing sequence (that is a k + 1 ≤ a k for all k )

4. Alternating Series Test Text suggests starting out by calculating If limit ≠ 0, you know it diverges If the limit = 0 Proceed to verify { a k } is a decreasing sequence Try it …

5. Using l'Hopital's Rule In checking for l'Hopital's rule may be useful Consider Find

6. Absolute Convergence Consider a series where the general terms vary in sign The alternation of the signs may or may not be any regular pattern If converges … so does This is called absolute convergence

7. Absolutely! Show that this alternating series converges absolutely Hint: recall rules about p-series

8. Conditional Convergence It is still possible that even though diverges … can still converge This is called conditional convergence Example – consider vs.

9. Generalized Ratio Test Given a k ≠ 0 for k ≥ 0 and where L is real or Then we know If L < 1, then converges absolutely If L > 1 or L infinite, the series diverges If L = 1, the test is inconclusive

10. Apply General Ratio Given the following alternating series Use generalized ratio test

11. Assignment Lesson 9.5 Page 636 Exercises 1 – 29 EOO