- About Midterm grades (for freshmen):
- I gave you the grade that I believe that you can earn (at least).
- You may not actually be earning that now! It depends, for example, on whether you do the correction to the exam described last time, etc.
- Let's make the test revisions due Friday, 11/7. Apologies to Corry!
- Reminder: Sehnert Lecture coming up!
- Note: I'll be replaced Friday and Monday by Don Krug, who'll be taking you through power series and more on Taylor series. The good news -- or bad news, depending on how you feel about them!;) -- is that we're going to be hearing more about the Taylor series polynomials, and these "infinite polynomials" in general.
- Assignment: To be handed in (due Thursday, 10/30): pp. 570-571, #7,
10, 11, 14, 17, 20-26, 28
-
Once again, our mission here is not necessarily the value of the
series, but may be simply knowledge of whether it converges or
not.
That being said, Theorem 2 below does give us some help in determining the value of (or at least bounds on the value of) the series.
- Series that converge absolutely, and
- Series that don't.
- Examples:
- #2, p. 570
- #6, p. 570
- #8, p. 570
- #18, p. 570:
- #27, p. 570
Is conditional convergence the end of commutativity?! Well, sort of!
- There are two main results (theorems) in section 11.5. They are
two tests for convergence of series:
This result says that eventually the ratio of successive terms is effectively constant,
: what kind of series looks like that? A geometric series:
This result says that eventually the absolute values of the terms are effectively equal to
: what kind of series looks like that? A geometric series!
No Wonder the results of the tests look exactly the same.... Too bad they didn't just use the same letter for the limits in both cases!
Notice that, once again, limits of sequences plays an important role! Series are just sums of sequences, after all; we're focused on how terms behave (root test), how successive terms behave (ratio test), or how partial sums behave.
- Examples:
- #1, p. 574
- #8, p. 575
- #13, p. 575
- #34, p. 575
- #36, p. 575
- Pick any....