Day 38 in Mat229

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Announcements
- Our quiz today will be over the divergence and integral tests.
  I asked you to try exercises #140-145, 152-155, 160-163, 170-171, 175-176, 190, 192 from that section. Any questions?
Today: finish the comparison tests, and start alternating series
- As we did for improper integrals, we find comparison with known creatures a useful approach. We look at several different approaches (including using improper integrals, via the Integral Test).
  Remember that it's all about the tail. In terms of convergence, we don't care about any handful of terms at the head of a series: \[ \sum_{k=1}^\infty a_k = \sum_{k=1}^n a_k + \sum_{k=n+1}^\infty a_k \] Convergence is about the tail: \(\displaystyle \sum_{k=n+1}^\infty a_k\)
- Materials for today:
  1. Comparison Test (.nb)
  2. Comparison Test (.pdf)
Next up: Alternating Series: Vol 2, Sec 5.5: Alternating Series
- These are among the friendliest series. I hope that you'll find these to be nice animals at the function zoo.
- The error formula is particularly easy for these series.
- Materials:
  1. Alternating Series (.nb)
  2. Alternating 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.

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