Testing Convergence At Endpoints

1. Testing Convergence At Endpoints Section 11.5

2. Integrate.

3. converges if , diverges if . p-series 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.

4. Does the series converge or diverge? converge converge diverge diverge converge diverge

5. 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.)

6. 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:

7. Example 1: Does converge? By the p-series test, we know it converges because 3/2 >1, but let’s show it using the Integral Test. First show that it is decreasing. Since the integral converges, the series must converge. (but not necessarily to 2.)

8. Does converge? Justify. First show that it is decreasing.

9. For , if then: if the series converges. if the series diverges. if the series may or may not converge. The Ratio Test

10. Use the Ratio Test to find whether or not the series converges. Since the Ratio test is inconclusive, now use the Integral Test.

11. Alternating Series

12. 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.

13. Here’s why: This series converges (by the Alternating Series Test.) This series diverges (harmonic series). So, the series is conditionallyconvergent.

14. This series converges (by the Alternating Series Test.) This series also converges (P-series test). So, the series is absolutelyconvergent.

15. Homework Page 513 #1-13 all

16. Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the absolute value of the first missing 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 (11.6).

17. Find an upper bound on the error if the sum of the first five terms is used as an approximation to the sum of the series. Since five terms are used, the error is at most the 6th term.

18. Find the smallest value of n for which the nth partial sum approximates the sum of the series within 0.005 (two decimal places).

19. Ratio Test Revisited Testing the Endpoints

20. For , if then: if the series converges. if the series diverges. if the series may or may not converge. The Ratio Test

21. What is the interval of convergence?

22. The interval of convergence is (2,8). The radius of convergence is . The series converges when

23. The interval of convergence is (2,8). Now check the endpoints. Diverges at 2 Diverges at 8 The interval of convergence is (2,8).

24. For what values of x does the Taylor series converge? Do the Ratio Test for the initial interval of convergence. Absolute convergence when: 1 Now test the endpoints:

25. When x=0: Diverges (harmonic series) When x=2: converges (alternating series) Complete interval of convergence:

26. Homework Page 514 #15-35 odd,36

27. Homework Wkst 8.3 #5-21, 33-39 odd

28. Since , converges to converges by the direct comparison test. Since converges absolutely, it converges. Ex. 4: We test for absolute convergence: