Day 44 in MAT229

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Announcements:
- It's the end of week 12, since we have some spring breaking to do. Get rested up for the sprint to the end of the semester!
- No lab tomorrow. I forgot to record today's session, but Prof. Wilkinson graciously shared his recording from earlier in the day.
Today: Power series as functions.
- We've seen that we can represent functions pretty well as Taylor polynomials. Now we create functions out of series of terms extending off into infinity.
  As an example, and one of the most important, consider 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).
  The questions we ask of these series are these:
  1. Where are they centered?
  2. What is the radius of convergence?
  3. What are their domains? (Often not the same as that of the given expression.)
  4. In particular, what happens on the boundary of the domain?
- Materials:
  - Functions as Power Series (.nb)
  - Functions as Power Series (.pdf)
  - Long talks you through
    - Functions as Power Series.
    - Calculus of Power Series.
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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