Day 47 in MAT229

Last time

Next time

Announcements:
- We've gotten to the middle of lucky week 13.
  Next weekend we have one more midterm exam, and still a fair amount of new material yet to encounter.
  We're finishing up series with Taylor series, and a reprise of Taylor Polynomials.
  Friday we begin parametric and polar equations.
- Your colleague Yimin found a problem with one of the IMath problems, in 11.9, and so I withdrew it. But I saved the work we did, so that you can check it out. It was #11.
Today:
- We've seen that we can represent functions pretty well as Taylor polynomials. But why stop at $n$?
  Because we need to stop somewhere, in most cases -- we don't have a way of coming up with an analytic answer to the sum all the way to $\infty$ (which we have in the case of a geometric series, for example).
- The Taylor remainder theorem provides us with an estimate of the error we're making in using that Taylor approximation, which is important -- it can guide us in choosing the degree of the polynomial we will use for an approximation.
- Materials:
  - Taylor Polynomials (.nb)
  - Taylor Polynomials (.pdf)
  - Long talks you through Taylor polynomials..
Links/Notes:
- 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.

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