1. The Comparison, Ratio, and Root Tests
Objective: Develop more convergence tests for series with nonnegative terms.

2. The Comparison Test
We will begin with a test that is useful in its own right and is also the building block for other important convergence tests. The underlying idea of this test is to use the known convergence or divergence of a series to deduce the convergence or divergence of another series.

3. The Comparison Test
We will begin with a test that is useful in its own right and is also the building block for other important convergence tests. The underlying idea of this test is to use the known convergence or divergence of a series to deduce the convergence or divergence of another series.

4. Using The Comparison Test
There are two steps required for using the comparison test to determine whether a series with positive terms converges.

5. Using The Comparison Test
There are two steps required for using the comparison test to determine whether a series with positive terms converges.
Guess at whether the series converges or diverges.

6. Using The Comparison Test
There are two steps required for using the comparison test to determine whether a series with positive terms converges.
Guess at whether the series converges or diverges.
Find a series that proves the guess to be correct. That is, if the guess is divergence, we must find a divergent series whose terms are “smaller” than the corresponding terms of If the guess is convergence, find a “bigger” series that converges.

7. The Comparison Test
The following are not formal theorems. In fact, we will not guarantee that they always work. However, they work often enough to be useful.

8. The Comparison Test
The following are not formal theorems. In fact, we will not guarantee that they always work. However, they work often enough to be useful.

9. Example 1
Use the comparison test to determine whether the following series converge or diverge.

10. Example 1
Use the comparison test to determine whether the following series converge or diverge.
The first looks like a divergent p-series (p = ½). We need to find a “smaller” divergent series.
The smaller diverges, so also diverges.

11. Example 1
Use the comparison test to determine whether the following series converge or diverge.
The second looks like a convergent p-series (p = 2). We need to find a “bigger” convergent series.
The larger converges, so also converges.

12. The Limit Comparison Test
Sometimes, it is very difficult to find a series to compare the given series to, making the comparison test useless. We will consider an alternative to the comparison test that is usually easier to apply.

13. The Limit Comparison Test
Sometimes, it is very difficult to find a test to compare the given series to, making the comparison test useless. We will consider an alternative to the comparison test that is usually easier to apply.

14. Example 2
Use the limit comparison test to determine whether the following series converge or diverge.

15. Example 2
Use the limit comparison test to determine whether the following series converge or diverge.
For the first series, we will say
Since and diverges, both series diverge.

16. Example 2
Use the limit comparison test to determine whether the following series converge or diverge.
For the second series, we will say
Since and converges, both series converge.

17. Example 2
Use the limit comparison test to determine whether the following series converge or diverge.
For the third series, we will say
Since and converges, both series converge.

18. The Ratio Test
The comparison test and the limit comparison test hinge on first making a guess about the convergence of a series and then finding something to compare it to. This is not always easy. The next test can often be used in this situation since it works exclusively with the terms of the given series- it requires neither an initial guess nor the discovery of a series for comparison.

19. The Ratio Test
The Ratio Test

20. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)

21. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(a)

22. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(b)

23. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(c)

24. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(d)

25. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(e)

26. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(e)

27. Example 3
Use the ratio test to determine whether the following series converge or diverge.
(a) (b) (c) (d) (e)
(e)

28. Root Test
In cases where it is difficult or inconvenient to find the limit required for the ratio test, the next test is sometimes useful.

29. Root Test
In cases where it is difficult or inconvenient to find the limit required for the ratio test, the next test is sometimes useful.

30. Example 4
Use the root test to determine whether the following series converge or diverge.
(a) (b)

31. Example 4
Use the root test to determine whether the following series converge or diverge.
(a) (b)
(a)

32. Example 4
Use the root test to determine whether the following series converge or diverge.
(a) (b)
(b)

33. Homework
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