- We're in week 9, an this week
we are talking about sequences and series.
- Note that we have a "spring break" day this week, so I'm going to
issue two days of new notes only -- nothing on Friday.
Friday you should relax and have some fun.
- 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 was a sophist, like a lawyer, who would take any case -- argue any position. He wrote several "paradoxes" that explored infinity, motion, etc., and tried to wrap his head around some very difficult ideas.
In one of his paradoxes, he showed that motion was impossible! Thank God it appears that he was wrong!
But he argued this way: to get to a wall, you have to first go half way; but to go half way, you have to go half of a half, or a quarter of the way; and before you go a quarter, you have to go an eighth; and so on, ad infinitum. So you can never get started, he argued!
Who can argue with that?:)
- Zeno understood, however, that it would certainly take you an
infinite amount of time to get to that wall (if you could, which is
impossible of course!). Because you'd have to half way to the wall, and
then half of a half, then half of a half of a half, etc. So it will
require an infinite number of actions, which must take an infinite
amount of time. So you're safe if someone tries to shoot an arrow at you:
He would have found it odd that we might write \[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} +\ldots + \frac{1}{2^n} + \ldots = 1 \ (\textrm{Ouch!}) \] and so assert that the arrow would, indeed, get there. So we want to investigate this idea of adding up an infinity of numbers and yet getting a finite sum....
- Infinite series (.nb)
- Infinite series (.pdf)
- Long talks you through the intro.
- Long talks you through geometric series to the end.
- Nothing to mention.