- We're starting lucky week 13; hopefully
you were able to relax a little over our "long
weekend", and get rested up for the sprint to the end
of the semester.
We have exactly four weeks of classes left, one more exam, and still a fair amount of new material yet to encounter.
We're finishing up series, with Taylor series (the extension of Taylor Polynomials); so that's both old and new.
Then it will be on to parametric and polar equations; extensions of these to three-space (curves in three dimensions); vectors in two and three dimensions; and finally lines and planes in space (3D).
- Your exams were graded
- Mean percentage: 76.75%
- SD: 13.24%
- Median: 77.5%
- Min: 50%
- Max: 93.75%
- The distribution (fairly normal), just so you know where you stand:
- key
- One observation: Generally we all need to be a little more verbose. One needs to justify answers. There were some who simply wrote "convergent" or "divergent", for example. We need a reason.
- We've seen that we can represent functions pretty well as Taylor
polynomials. But why stop at $n$?
- Power series suggest that we can just keep on going with those
Taylor polynomials, all the way to infinity -- and it's
true. And the strategy will be the same (with some of the new
questions asked of power series, which weren't an issue with
those finite polynomials):
- Where should we center this infinite polynomial?
- For coefficients, we'll choose them to match derivatives of the function at that center.
- Then for what values will this particular power series be valid?
- And, in particular, what happens on the boundary of the domain?
As an example, and one of the most important, we've already seen that the "geometric series" function \[ \frac{1}{1-x}=\sum_{n=0}^\infty x^n \] (where $x$ plays the role that we've been having $r$ play).
- Materials:
- Taylor Series (.nb)
- Taylor Series (.pdf)
- Long talks you through Taylor Series.
- The Riemann
series theorem: if an infinite series of real numbers is
conditionally convergent, then its terms can be arranged in a
permutation so that the new series converges to an arbitrary real
number, or diverges.