Day 42 in Mat229

Last time

Next time

Announcements
- Your quiz today will be over comparison tests and the AST.
Last time:
1. we reviewed the AST, and the associated remainder estimate.
2. we discussed absolute and conditional convergence. This provides us a tool for dealing with negative terms in a series, even when they're not alternating.
3. Absolutely convergent series (those whose partner series of the absolute values of their terms also converge) behave a lot like finite sums.
4. The coolest thing about conditionally convergent series is that by rearranging the terms and adding them up in a different order, you can get different results. This might seem disturbing -- feel welcome to feel uncomfortable!
  Infinity is sometimes disturbing.
Today:
- We continue with power series, takes us in a different, and very important, direction. Last time we explored using a workhorse of a "power series" explored in this section. \[ \sum_{k=0}^\infty x^k = \frac{1}{1-x} \] which is convergent when \(|x|<1\).
  Our materials for power series:
  1. Last time we were working out some examples from the text, in this PowerSeries.nb file (PowerSeries.pdf) -- from the text's section on Power series (Vol 2, Sec 6.1: Power Series and Functions)
  2. (Power_series.nb)
  3. (Power_series.pdf)
- So far we've worked with power series, but don't know yet how to make them from scratch. Today we go into the power series kitchen to see how that's done.
  Our materials for Taylor series:
  1. Taylor_series.nb
  2. Taylor_series.pdf

Links:

Prof. Roger Zarnowski's cool series tools handout
Wolfram Alpha
Mathematica on-line is an option, if you are at a computer without Mathematica installed.

Website maintained by Andy Long. Comments appreciated.