Day 39 in MAT229

Today:

• Announcements:
  • Weekly Assignment 6 is ... graded! Hooray!
    1. We are interested in \[ \sum_{k=1}^\infty a_k \] Some of you show $\sum_{k=1}^\infty a_k$ converges by comparison with $\sum_{k=1}^\infty b_k$, use $\sum_{k=1}^\infty b_k$ to find $n$ for an error bound, and then produce \[ S_n=\sum_{k=1}^n b_k \] Whoops! We weren't interested in $b$ -- you dragged that in to show convergence/determine $n$ for an error bound. We want \[ S_n=\sum_{k=1}^n a_k \]

    2. Some of you are being too casual in your comparisons. You're not at the stage yet when you can work entirely off thought bubbles (although I am loving your thought bubbles!). I need details.

       So if you claim limit comparison, you must do a limit; if you claim comparison, you must find $b_k \ge a_k$ (or $b_k \le a_k$, whichever is appropriate), and you must show that the inequality is, indeed, satified.

       If you're going to use results related to alternating series, mention AST or the like (and justify -- e.g. terms must alternate, be decreasing and headed for 0).

    3. What's wrong with this sum? \[ \sum_{k=0}^\infty \frac{1}{k^3} \]

    4. You can get different answers for some problems, depending on what you choose for comparison. For example, on 1.c some of you chose $\frac{1}{k^3}$ and some of you chose $\frac{1}{k^2}$ and some of you chose $\frac{1}{2k^3}$. Each will produce a different $n$, but ALL should produce an estimate to within .0001.

       By the way, all of these need be larger only eventually.

    5. Some of you thought that 20000 terms seemed way too high! No such thing. The point of problem 2 is that we have two ways to approximate $\pi$ to within .0001: one requires 20000 terms, and another which requires only 8 terms. Which should we use?
    James, 1a; Elise, 1b; Abby, 1c; Andrew, 2a; Patrick, 2b; Semin, 2c

  • Weekly #7 is due next Tuesday.

  • Early head's up: exam 3 is Tuesday, 4/9.

• Today:
  • We wrap up Sect. 11.9: Functions as power series (pdf, pdf summary)