1. 9.6 The Ratio & Root Tests Objectives:
Use the Ratio Test for series convergence
Use the Root Test for series convergence
Review tests for convergence and divergence of infinite series
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2. Important Idea
Theorem-Ratio Test ( test for absolute convergence) in 3 parts:
Let have non-zero terms:
converges absolutely if

3. Important Idea
Theorem-Ratio Test ( test for absolute convergence)
diverges if or

4. Important Idea
Theorem-Ratio Test ( test for absolute convergence)
3. The Ratio Test is inconclusive if
The Ratio Test works best with series involving factorials or exponentials

5. Example
Determine the convergence or divergence of:
P590, ex1

6. Try This Determine the convergence or divergence of: Diverges
P590, Ex 2b
Diverges

7. Try This Determine the convergence or divergence using the Ratio Test:
P591,ex 3
Inconclusive

8. Important Idea Theorem-The Root Test (in 3 parts):
Let have non-zero terms:
converges absolutely if

9. Important Idea
Theorem-The Root Test (in 3 parts):
diverges if or

10. Important Idea Theorem-The Root Test (in 3 parts):
3. The Root Test is inconclusive if
The Root Test works best with series involving nth powers

11. Example Determine the convergence or divergence of:
P592 ,ex 4-converges

12. with Mrs. DeBelina
There are 10 tests for convergence and divergence of a series.
So many tests; so little time. How do I decide?

13. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
1. nth Term Test for divergence:
divergence
and you are done.

14. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
2. Determine if the series is one of the special types:
a) geometric b) p c) harmonic d) telescoping e) alternating

15. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
a) geometric series converges to if
otherwise diverges.

16. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
b) p series converges if p>1
,otherwise diverges

17. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
c) harmonic series is a special case of the p series
where p=1. It diverges. The general form is

18. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
d) telescoping series
converges to b1-L if

19. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
e) alternating series
converges if
and

20. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
3. Direct Comparison with a geometric or p-series known to converge. If and
converges, then
converges.

21. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
4. Limit Comparison with a geometric or p-series known
to converge. If
and
converges, then
converges.

22. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
5. Ratio Test for exponentials or factorials.
converges
if L<1 and diverges if L>1
where

23. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
6. Root Test for nth powers.
converges
if L<1 and diverges if L>1
where

24. Important Idea
Applying the tests in this order, as applicable, means you don’t duplicate work:
7. Integral Test (f must be continuous, positive and decreasing).
and
both converge or diverge.

25. Lesson Close
A summary of the convergence/divergence tests for infinite series is found on page 644 of your text.