Day 43 in MAT229

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Announcements:
- It's week 12.
- It's also our only lab day this week: so I hope that everyone comes who wants to! Zoom this lab.
- I'm sorry I've not made any progress on exams. Too many other things going on. Hopefully soon....
- I learned something very interesting over the weekend, at the meetings of the Kentucky section of the Mathematical Association of America: that \[ 1 + 2 + 3 + 4 + \ldots = -\frac{1}{12} \] This result, due to Ramanujan (1887-1920). So that's how they did math in the old days...:)
  Of course they may have been doing things in somewhat different fashion than we imagine; and this is related to Grandi's series (your classmate Estelon Eastham mentioned this the other day; Grandi was born in 1671): \[ A = 1 - 1 + 1 - 1 + 1 - 1 + \ldots \] Of course the series is divergent by the divergence test! But it doesn't mean that it's useless....
  I have your classmate Caeser Bao in another course, and in it he used the Grandi series to prove that $0 = 1$:
```
			0=1
			0=0+0+0+0+0+0+...
			0=(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
			0=1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
			0=1+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0...
			0=1
			
```
  I was quite surprised, of course, because I always thought that $0=2$:
```
			0=2
			0=0+0+0+0+0+0+...
			0=(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
			0=1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
			0=1+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+...
			0=1+1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+...
			0=1+1+0+0+0+0+0+0+0+0+0+0+0+0+0+0...
			0=2
			
```
- Clearly things are not what they seem when you try to manipulate infinite sums!
Today: Lab 12
- The next topic we consider, power series, takes us in a different, and very important, direction. One of the most important questions will be this: "What values of $x$ are computable -- that is, for what values of $x$ does a power series converge?"
  We will often turn to the tests we consider today for the answer.
  
  Computer lab 12 (nb) (pdf):
Links/Notes:
- The Riemann series theorem: if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges.

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