Hi there, folks of 21C discussion,

IF YOU MISSED YESTERDAY...

There was no quiz this week, but we begin all the remaining sessions with a 15 minute quiz at the beginning (3:10 sharp).  It is in your best interest to make it as on-time as possible.  I will communicate announcements: in class, using this e-mail list, and on the discussion mini-webpage at:
http://www.math.ucdavis.edu/~ekim/teaching/0708/spring/21C/

If you missed yesterday, please review the weblink.  I also handed out a 'Section Syllabus/Information' sheet (posted on the webpage).  Also, please fill out the Questionnaire (also posted).

OVERVIEW OF CHAPTER 11

Showing the correct work for the convergence/divergence of sequences (11.1) and series (11.2-11.6) is very different from the computational nature of 21A and 21B.  The burden will be on you in quizzes and exams (so, you should try to practice the same on your HW scratch work) to justify exactly WHY a sequence or a series converges or diverges by stating BY NAME which test you use.

ADDITIONAL RESOURCES

I've posted two e-handouts (posted/dated April 4th on the webpage)... one for sequences and one for series.  These are not exhaustive, but I hope they are useful for review. Don't directly use sequence tests on series, and don't use series tests on sequences. (However, some series tests will make you examine some OTHER sequence related to the series. To study those sequences, you should use the sequence rules.)

I also posted an optional worksheet (Series Intuition Builder). I can't require it, but I wrote it to address the main pitfalls from the last time I taught 21C. The unfortunate thing is that you'll probably cover one or two tests per lecture. Perhaps the best way to use this sheet is one row at-a-time: For instance, right after the lecture that covers the Geometric Series Test, try following the instructions for each square in the first row of the grid.

After about week 3, you'll have the whole thing filled out. Then, print it out again, and try it one column-at-a-time, instead of one row-at-a-time. This will simulate better what happens in a quiz or midterm. I realize you have other classes, but do as much of this worksheet as you can after each lecture. We'll try and find time to go over this in discussion. As you know, in general you'll get the most out of discussion by having tried both the required and suggested HW before class.

SUMMARY OF YESTERDAY

Unfortunately, the only correct definition of 'convergence of sequences' is given by a lengthy and hard-to-grasp sentence (see box on p733).  Intuitively, a sequence converges if, in the long run, the sequence narrows in on some value.  When graphed on the Cartesian plane, we ask: as we run up the 'x' (really the 'n') value, or rather, as we move to the right, does the y-coordinate (or height) settle down on one number?

None of the sequences in problems 13-22 settle down on one number.  By drawing a picture (a graph) of the sequence, we get intuition on the question of convergence.  We drew out the terms of problem 32 to say the sequence diverges (and the specific kind if divergence here was oscillation) because it settles on two numbers, and not one.  A student asked a very good question if the (-1)^n would always cause the oscillation that gives divergence.  We examined a_n = (-1)^n * 2/n and noted this sequence converges to 0.  In conclusion, there is no uniform rule regarding  sequences whose terms alternate in sign.

For 27, we used L'Hopital's Rule.  If you feel rusty on this material, review section 4.6 in the book.  In 33, we used the Product Rule (which is one of the Laws of Limits given in the box on p735).  There are several other rules, which we did not cover.  For example, we did not do a problem using the Sandwich Rule.

Have a great weekend,
Eddie