Day 42 in MAT229

Last time

Next time

Announcements:
- It's week 12.
- I hope that exam went okay -- I haven't had a chance to look at them yet.
- This week we have a couple of days off for "Late Spring Break" (to coincide with Easter, I guess). So there will be no lab on Thursday, and no new material on Friday.
  Anyone is welcome to come to lab on Tuesday, and it will be zoomed and recorded (so long as I remember; and you can help!).
Today: power series
- The next topic we consider, power series, takes us in a different, and very important, direction:
  
  What if we replace the constants in the series with functions?
- It's easy enough to think of an example like this: what if I change the size 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: \[ \sum_{k=1}^\infty \left(\frac{1}{2}\right)^k = \sum_{k=1}^\infty \frac{1}{2^k} \]
- Materials:
  - Power Series (.nb)
  - Power Series (.pdf)
  - Long talks you through Power Series.
Links/Notes:
- Nothing to mention.

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