Day 41 in Mat229

Last time

Next time

Announcements
- Your quiz on Thursday will be over comparison tests and the AST.
- Please skim Vol 2, Sec 6.2: Properties of Power Series and read Vol 2, Sec 6.3: Taylor and Maclaurin Series for next time.
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 discussing absolute convergence, and the root and ratio tests. We then apply them to "power series".
- There are two more tests for convergence of series that we need to explore:
  1. Ratio test: Let $\{a_k\}$ be a sequence and assume that the following limit exists: \[ r=\lim_{k\to \infty}\left|\frac{a_{k+1}}{a_k}\right| \]
    1. if $r<1$, then the series $\sum a_k$ converges absolutely.
    2. if $r>1$, then the series $\sum a_k$ diverges.
    3. if $r=1$, the test is inconclusive.
    This result says that eventually the ratio of successive terms is effectively constant, r: the terms of the sequence approach a "common ratio" as $k\to\infty$. What kind of series looks like that? A geometric series:
    \[ \sum_{k=0}^{\infty}{c}r^k \]
    The ratio test is effectively a self-referrential comparison test: we compare terms of $a_n$ with other terms $a_{n+1}$ (rather than with some other series).
  2. Root test: Let $\{a_k\}$ be a sequence and assume that the following limit exists: \[ r=\lim_{k\to \infty}\sqrt[k]{|a_k|} \]
    1. if $r<1$, then the series $\sum a_k$ converges absolutely.
    2. if $r>1$, then the series $\sum a_k$ diverges.
    3. if $r=1$, the test is inconclusive.
    This result says that eventually the absolute values of the terms are effectively equal to $|a_k|={r^k}$: what kind of series looks like that? A geometric series!
    Notice that, once again, limits of sequences plays an important role. Series are just sums of sequences, after all; we're focused on how terms behave (root test), how successive terms behave (ratio test), or how partial sums behave.
  Our first materials for today:
  1. (Absolute.nb)
  2. (Absolute.pdf)
- The next topic we consider, power series, takes us in a different, and very important, direction:
  
  What if we replace the constants in these series with functions?
  - Remember Zeno, the guy who proved that he couldn't even get across a room of unit length, of length one?
    Motion is impossible: Before he could take half a step he had to take a quarter step, and before he took a quarter step he had to take an eighth step, and so on. So he could never get started....
    Motion doesn't get you far: Then he decided that if he ever did take a half step, and then a quarter step, and so on, he'd never reach the far wall, because he'd have an infinite number of steps to take.
    What if I change the length of the room, and then run Zeno's experiment? I can take an $x/2$ step, then $x/4$, then $x/8$, etc. How far will I have gone if I add up the entire infinite series?
    \[ \sum_{k=1}^\infty\frac{x}{2^k} = x\sum_{k=1}^\infty\frac{1}{2^k} = x \]
    No big deal, right?
  - Now what if we take steps that are more complicated functions of $x$? E.g. what if we take steps that are polynomial functions of the step number, e.g.
    \[ \sum_{k=1}^\infty x^k \] Well actually, if we let $x=\frac{1}{2}$, this is Zeno's problem: \[ \sum_{k=1}^\infty \left(\frac{1}{2}\right)^k = \sum_{k=1}^\infty \frac{1}{2^k} = 1 \] More generally, this is a workhorse of a "power series" explored in our new section for today. \[ \sum_{k=0}^\infty x^k = \frac{1}{1-x} \] which is convergest when $|x|<1$.
  - Our next materials:
    1. PowerSeries.nb -- examples 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)
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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