The ratio and root test

1. The ratio and root test

2. 1 There is a positive number R such that the series diverges for but converges for . The series converges for every x. ( ) 2 3 The series converges for at and diverges everywhere else. ( ) Recall: There are three possibilities for power series convergence. The series converges over some finite interval: (the interval of convergence). The series may or may not converge at the endpoints of the interval. (As in the previous example.) The number R is the radius of convergence.

3. where r = common ratio between terms When , the series converges. Ratio Technique We have learned that the partial sum of a geometric series is given by:

4. For , if then: if the series converges. if the series diverges. if the series may or may not converge. Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge.

5. If is a series with positive terms and then: The series converges if . The series diverges if . The test is inconclusive if . The Ratio Test:

6. Determine if the series converges

8. Does the series converge or diverge?

9. Does the series converge or diverge? Series diverges

10. If is a series with positive terms and then: The series converges if . The series diverges if . The test is inconclusive if . Nth Root Test: Note that the rules are the same as for the Ratio Test.

11. Helpful tip When using the root test we often run into the limit nth root of n as n approaches ∞ which is 1 (We prove this at the end of the slide show)

12. example: ?

13. it converges example: ?

14. it diverges another example:

15. Tests we know so far: Try this test first nth term test (for divergence only) Then try these Special series: Geometric, Alternating, P series, Telescoping General tests: Ratio Test Direct comparison test, Limit comparison test, Root test Integral test, Absolute convergence test (to be used with another test)

16. Homework P 647 13-31 odd, 51-65 odd 87-92 all How can you measure the quality of a bathroom? Use a p-series test By Mr. Whitehead

17. formula #104 formula #103 Indeterminate, so we use L’Hôpital’s Rule

18. Extra example of ratio testDoes the series converge or diverge?

19. Extra example of the ratio test Does the series converge or diverge?