9.5 Testing Convergence at Endpoints

1. Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington 9.5 Testing Convergence at Endpoints Petrified Forest National Park, Arizona

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

3. This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series.

4. 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.

5. example: ?

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

7. it converges example: ?

8. it diverges another example:

9. The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge. Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to:

10. Example 1: Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.)

11. converges if , diverges if . p-series Test We could show this with the integral test. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

12. It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series. the harmonic series: diverges. (It is a p-series with p=1.)

13. If , then both and converge or both diverge. If , then converges if converges. If , then diverges if diverges. Limit Comparison Test If and for all (N a positive integer)

14. Since diverges, the series diverges. Example 3a: When n is large, the function behaves like: harmonic series

15. Since converges, the series converges. Example 3b: When n is large, the function behaves like: geometric series

16. Alternating Series Test If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series The signs of the terms alternate. Good news! example: This series converges (by the Alternating Series Test.) This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent.

17. Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. Since each term of a convergent alternating series moves the partial sum a little closer to the limit: This is a good tool to remember, because it is easier than the LaGrange Error Bound.

18. F3 4 There is a flow chart on page 505 that might be helpful for deciding in what order to do which test. Mostly this just takes practice. To do summations on the TI-89: becomes becomes

19. ENTER ENTER GRAPH Y= WINDOW To graph the partial sums, we can use sequence mode. MODE Graph……. 4

20. ENTER ENTER GRAPH Y= WINDOW Table To graph the partial sums, we can use sequence mode. MODE Graph……. 4

21. ENTER ENTER GRAPH Y= WINDOW Table To graph the partial sums, we can use sequence mode. MODE Graph……. 4 p