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Section 11.3 Part II The Comparison Tests

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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?


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