1. Section 10.6 Comparison Tests

2. 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.

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

4. Direct Comparison Test Direct comparison fails, but it still “feels” like the series should converge.

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

6. 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.

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

8. Ex: #6 (read the directions) 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.

9. Ex: #12 The series converges by comparison. This seems too easy …what’s the catch?

10. Ex: #22 With the exception of the first term, This series diverges by direct comparison to the Harmonic Series.

11. 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 an LCT. The series diverges by the LCT.