1. Section 11.3 Part II The Comparison Tests
MAT 1236 Calculus III
Section 11.3 Part II
The Comparison Tests

2. HW …. WebAssign 11.3 Part II Write out your solutions carefully
Quiz: 11.2 II, 11.3

3. Come talk to me! If you are not 100% sure about...
I am in my office MTRF from 3 to 4pm
This is the first year in calculus III that very few come to ask me questions...

4. Reminder to Wai Do the first 3 examples on the board without erasing
Overwrite on the first 3 examples to save time

5. Preview Comparison Test Limit Comparison Test
Only work for series with positive terms
Compare with standard series
The nature of the comparison is
term-by-term

6. Examples
Comp. Test Limit Comp. Test

7. Examples
Comp. Test Limit Comp. Test

8. How does it diverge?

9. The Comparison Test Suppose 𝑎𝑛 𝑏𝑛>0 for all 𝑛.
If 𝑎𝑛 is convergent then 𝑏𝑛 is also convergent
If 𝑏𝑛 is divergent then 𝑎𝑛 is also divergent

10. Be Careful!!!!! Suppose 𝑎𝑛 𝑏𝑛>0 for all 𝑛.
If 𝑎𝑛 is convergent then 𝑏𝑛 is also convergent
However,
If 𝑏𝑛 is convergent then there is no conclusion for 𝑎𝑛

11. Example 1

12. Series to compare with
Geometric Series
Harmonic Series
𝑝-series

13. Remarks For convenience, we will call the following series a p-series.
It has the same convergence as

14. Series to compare with
Geometric Series
Harmonic Series
𝑝-series

15. Example 1 Write down the general terms of the two series
Write down the comparison and range
Find the convergence of the comparison series
Make the conclusion by quoting the name of the comparison test

16. Common Mistake/Misconception
Comparing the series instead of comparing the terms
STOP
The series are not comparable unless you first show that they are both convergent.
The comparison test is based on the comparison of general terms.

17. Examples
Comp. Test Limit Comp. Test

18. Example 2
Remark: Need to justify the inequality if not obvious.

19. Examples
Comp. Test Limit Comp. Test

20. Example 3

21. Examples
Comp. Test Limit Comp. Test ?

22. The Limit Comparison Test (L.C.T.)
Suppose 𝑎𝑛, 𝑏𝑛>0 If
then both 𝑎𝑛, 𝑏𝑛 converge or diverge

23. The Limit Comparison Test (L.C.T.)
Suppose 𝑎𝑛, 𝑏𝑛>0 If
then both 𝑎𝑛, 𝑏𝑛 converge or diverge

24. Example 4

25. Examples
Comp. Test Limit Comp. Test ?

26. Example 5

27. Examples
Comp. Test Limit Comp. Test ? ?

28. Example 6

29. Examples
Comp. Test Limit Comp. Test ? ?

30. HW #13 Will requires some thoughts.
Do not just ask the tutor (or the internet) for the answers.
Doing challenging problems are part of the education/learning process.
Solution will be posted tomorrow.

31. Challenge #1..On your own...
Can you figure out another method for HW #13?

32. Challenge #2..On your own...
Is it possible to use the comparison test to get a result for example 3?