1. Infinite Series 9 Copyright © Cengage Learning. All rights reserved.

2. Comparisons of Series Copyright © Cengage Learning. All rights reserved. 9.4

3. Use the Direct Comparison Test to determine whether a series converges or diverges. Use the Limit Comparison Test to determine whether a series converges or diverges. Objectives

4. What is the most important point of the previous lessons? The terms must go to zero “fast” enough. too slow - diverges fast enough - converges The next few lessons will explore a myriad of methods that test how “quickly” the terms are going to zero and set criteria to determine whether or not the series converges.

5. In this lesson we will compare new series to series we already know about. Known Convergent SeriesKnown Divergent Series

6. Direct Comparison Test Direct comparison fails, but it still “feels” like the series should converge. Let f and g be continuous on with for all, then: 1 converges if converges. 2 diverges if diverges.

7. Limit Comparison Test The series converges by comparison to a convergent p-series. then either both series converge or both diverge.

8. The limit comparison test computes the ratio being approached by the terms of the two series. If this ratio is some positive real number, then the terms are more or less proportional. So then the “speeds” at which the 2 series are approaching zero are proportional. Their sums (if they exist) are almost proportional. They are both therefore either approaching zero fast enough for convergence or not.

9. Some handy algebraic comparisons to keep in mind: Some handy limits to keep in mind: You must understand… not memorize! Whatever.

10. Ex: #1 Determine the convergence or divergence: Choosing a familiar series that is similar is the first and most important step! The “10” can be moved to the front of the summation.

11. Ex: #2 The series converges by comparison. This seems too easy …what’s the catch?

12. Ex: #3 With the exception of the first term, This series diverges by direct comparison to the Harmonic Series.

13. Ex: #4: The maximum value of so: on Since converges, converges.

14. Ex: #5 Show that the following general harmonic series diverges. Solution: By comparison with you have Because this limit is greater than 0, you can conclude from the Limit Comparison Test that the given series diverges.

15. Here’s a trickier one: However, less than a divergent tells us nothing. For this series we have to go back to the integral test with a much more similar function. Now perform n LCT. The series diverges by the LCT.

16. Day 1: Pg. 628 1-35 odd Day 2: Pg. 628 4-36 even Day 3: MMM pg. 202-203