1. Alternating Series; Absolute and Conditional Convergence

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
{ ak } is a decreasing sequence (that is ak + 1 ≤ ak 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 { ak } 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
ak ≠ 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 8.6
Page 542
Exercises 5 – 29 EOO