Series with Positive terms: tests for Convergence, Pt. 1

1. Series with Positive terms: tests for Convergence, Pt. 1 The comparison test, the limit comparison test, and the integral test.

2. In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere. Comparing series. . . Consider two series , with for all k.

3. Comparing series. . . Consider two series , with for all k. How are these related in terms of convergence or divergence? Note that: What does this tell us?

4. Where does the fact that the terms are non-negative come in? Comparing series. . . Consider two series , with for all k. Note that: What does this tell us?

5. Since for all positive integers k. Then So the sequence of partial sums is . . . Series with positive terms. . . Non-decreasing Bounded above Geometric

6. Back to our previous scenario. . . Consider two series , with for all k. Suppose that the series converges

7. So the sequence of partial sums is . . . Suppose that the series converges Non-decreasing Bounded above Geometric

8. Suppose that the sequence is non-decreasing and bounded above by a number A. That is, . . . Then the series converges to some value that is smaller than or equal to A. A variant of a familiar theorem Theorem 3 on page 553 of OZ

9. So the sequence of partial sums is . . . Suppose that the series diverges For all n we still have Non-decreasing Bounded below Unbounded

10. This gives us. . . The Comparison Test: Suppose we have two series , with for all positive integers k. • If converges, so does , and • If diverges, so does .

11. A related test. . . This test is not in the book! There is a test that is closely related to the comparison test, but is generally easier to apply. . . It is called the Limit Comparison Test

12. (One case of…) The Limit Comparison Test Limit Comparison Test:Consider two series with , each with positive terms. • If , then are either both convergent or both divergent. Why does this work?

13. (Hand waving) Answer: • Because if Then for “large” n, ak  t bk. This means that “in the long run”

14. The Integral Test Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . . y = a(x) Now we add some enlightening pieces to our diagram….

15. y = a(x) The Integral Test Suppose that we have a sequence {ak} and we associate it with a continuous function y = a(x), as we did a few days ago. . . Look at the graph. . . What do you see?

16. So y = a(x) The Integral Test converges diverges If the integral so does the series.

17. The Integral Test y = a(x) Now look at this graph. . . What do you see?

18. So y = a(x) The Integral Test Why 2? converges diverges If the integral so does the series.

19. The Integral Test The Integral Test: Suppose for all x 1, the function a(x) is continuous, positive, and decreasing. Consider the series and the integral . If the integral converges, then so does the series. If the integral diverges, then so does the series.

20. The Integral Test The Integral Test: Suppose for all x 1, the function a(x) is continuous, positive, and decreasing. Consider the series and the integral . If the integral converges, then so does the series. If the integral diverges, then so does the series. Where do “positive and decreasing” come in?