The Direct Comparison Test: The idea here can be summarized as "Over" and "Under".
A series with terms that are greater than (Over) the terms of a divergent series, must diverge also.
A series with terms that are less than (Under) than the terms of a convergent series, must converge also.
Note: This reasoning applies only to series with positive terms; otherwise, "Under" could be a case where -1000 is "under" +1/2, and uh oh, the floor dropped out! So we have to make sure we apply this in legitimate situations...no sink holes (negative terms) allowed!.
Videos:
Spot Check: Practice!
Note: Some of the practice activities on Khan Academy are useful for our course and some aren't. If I insert one of the "Practice" activities in here, it's something I think would be really helpful in building your understanding. It isn't meant to replace the homework, rather it's just very focused practice with a concept.
The Limit Comparison Test: Okay, now we're going to start using more intuition about whether we think a series converges or not without having to fuss with the details of strict inequalities. This will get you thinking more in terms of the rate at which the terms go to zero (remember, that's a NECESSARY condition for convergence). We then can back up our intuitive guesses with a formal Limit Comparison Test to prove our intuition was correct.
Videos:
Spot Check: Practice!
Assigned Problems from Textbook
Note: This is a scaled-down set of homework problems as compared to the original assignment sheet.
|
1st Edition Section 9.5 pg. 668 27, 28, 29, 31, 33, 34, 35, 38, 39 3rd Edition Section 10.5 pg. 687 9, 10, 11, 17, 23, 24, 25, 30, 37abc |