1. Tests for Convergence, Pt. 1 The comparison test, the limit comparison test, and the integral test.

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

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

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

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

7. The two “ingredients” together... and converges Terms of a series are non-negative Partial sums are non-decreasing Partial sums of are bounded above.

8. A variant of a familiar theorem 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. Theorem 3 on page 553 of OZ

9. 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.

10. A related test... 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 This test is not in the book!

11. (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?

12. (Hand waving) Answer: Because if Then for “large” n, a k  t b k.This means that “in the long run” the partial sums behave similarly in terms of convergence or divergence.

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

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

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

16. The Integral Test If the integral so does the series. So converges diverges Why 2?

17. 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.