1. 9.4 Comparison of Series 9.5 Alternating Series

2. The Direct Comparison Test The Boring Book Definition: (BOOO!!!) Part 1 The series diverges if there exists another series such that b n < a n and b n diverges, then the series diverges. Part 2 The series converges if there exists another series such that a n <b n and b n converges, then the series converges.

3. The Analogy (Yippee!!!) Ok... Imagine for a moment that you have a MAGIC BOX (ooh! aah!) This magic box has the ability to float in the middle of the air! (wow!)

4. Where will the magic box go? You decide that want to push this magic box somewhere... either on the ceiling or on the floor...

5. Push it to the floor If you want to push it to the floor, your hands need to be ON TOP of the box, and you need to push downward

6. Push it to the ceiling If you want to push it to the ceiling, you need to put your hands underneath the box, and then push upward

7. Going nowhere... If your hands are ON TOP of the magic box, and you push UPWARD, What happens to the box? NOTHING!

8. Going nowhere, again! If your hands are UNDERNEATH the magic box, and you push DOWNWARD, What happens to the box? NOTHING!

9. Put it all together... For the Direct Comparison Test, you can think of the “magic box” as the series which you want to find out if it converges or diverges. You can think of your hands as the other series Pushing downward means your converges Pushing upward means your diverges

10. The Direct Comparison Test Part 1 (rewritten) If you want to show that converges: This is like trying to push the magic box down to the floor. You need to have two things 1. You need to have your hands (b n ) on top of the “magic box” (a n ) Mathematically, this means that b n > a n 2. You need to have your hands ( ) push downward Mathematically, this means has to converge If you can find b n that does this, then converges

11. The Direct Comparison Test Part 2 (rewritten) If you want to show that diverges, This is like trying to push the magic box up to the ceiling. You need to have two things 1. You need to have your hands (b n ) underneath the “magic box” (a n ) Mathematically, this means that b n < a n 2. You need to have your hands ( ) push upward Mathematically, this means has to diverge If you can find a b n that does this, then diverges

12. The Direct Comparison Test Limitations Remember that if you have your hands ON TOP of the magic box and you push upward, nothing happens... Similarly: If you have a b n > a n and diverges, then nothing happens... you don’t prove anything If you have your hands UNDERNEATH the magic box and you push downward, nothing happens... Similarly: If You have a b n > a n and diverges, then nothing happens... you don’t prove anything

13. Example: Let’s try to show that converges. Let’s place in our magic box If you want to show that converges, then we need to put our hands on top of it and push downward...

14. Example: continued... Let’s try to choose something SIMPLE that (we already know) converges... Let’s choose This new series that we chose will become our “hands”. So now, we have to figure out if our “hands” go ON TOP or UNDERNEATH the MAGIC BOX. If we plot the graphs of and We can easily see that the graph of is on top of

15. Example: continued... Since the graph of our “hands” is on top of the graph of “the magic box, this means that we can place “our hands” on top of the magic box And since we know ahead of time that converges, this means that we have everything we need to push the magic box downward... thus proving that converges!

16. Example 2

17. Example 3

18. Limit Comparison Test

19. Examples

21. Works Great When You Have a Mess! Compare “messy” algebraic series with p-series

22. Examples

24. Alternating Series Test

25. Example

29. Alternating Series Remainder