Day 30 in Mat229

• Our quiz today will be over improper integrals. Any questions?

• Our lab tomorrow will include a discussion of Newton's method, which is described in your sequences section.

• Here are some problems from the last three sections:
  • Improper integrals: 347-350, 353-354, 355-363 (odd), 372-373, 394-395.
  • Sequences: 2, 3, 5, 8, 13, 21 (why did I choose those numbers?:); 23-26, 27-34, 46-49, 59, 60
  • Series: 67-69, 71-73, 79-80, 89-91, 93-94, 105, 116
  You might want to work those in preparation for your second exam, which is the Thursday following Spring Break. It will feature the three sections above, as well as
  • Numerical integration

and just glimpsed series. We talked briefly at the end of class about Zeno's paradoxes, and about the fundamental idea of adding up an infinite number of things.

As I mentioned, the trick to evaluating a series (an infinite sum) will be to consider a related sequence: the partial sums. That is, we are going to add up 1 terms, 2 terms, 3 terms, etc. -- all perfectly legitimate, because these are finite sums -- and then see if that sequence converges.

If it does, we'll say that the series has a value equal to the limit of the sequence.

• Series are analogous to improper integrals, where the domain of the integration trots off to $\infty$. We're trying to add up an infinitely large number of things, and hoping that it will give a finite number.

  Naturally, this sounded a little odd to folks for a long time. It may even sound odd to you, and that's okay!

  Zeno discussed a number of paradoxes, such as

  1. why you needn't fear an arrow: It would never get to you! Hooray!

  2. why Achilles couldn't even catch a tortoise:

• Let's turn to our handout for this section (and here's the Mathematica file)