From The Everything Seminar: Sum Divergent Series, I". Matt suggests that
1 + 2 + 4 + 8 + 16 + 32 + ... = -1
and explains why. He promises more in this vein. As many of you might see, this formula comes from taking the well-known sum of a geometric series,
1 + x + x2 + x3 + ... = (1-x)-1
and letting x=2. This is true in the 2-adic numbers. It is also true in certain sorts of computer architectures; read item 154 of HAKMEM to find out which ones. (Read the rest of HAKMEM while you're at it. Well, except for the hardware part at the end. Hardware is stupid.)
Something that comes to mind when I think about series like this is that the Riemann zeta function is often defined in popular books (such as the various recent books on the Riemann hypothesis) as
ζ(s) = 1-s + 2-s + 3-s + 4-s + ...
and this series only converges when the real part of s is greater than 1. These books then go on to talk about how all the zeroes of the zeta function are either negative even integers (the "trivial zeroes") or are believed to have real part 1/2 (this last part being the Riemann hypothesis). Yet as far as I remember, they never point out that the definition above doesn't properly make sense for any of the values of s just mentioned; you have to analytically continue the function from the half-plane where it's already defined. There's a unique way to do this, so it's not a big problem, but it's a little bit annoying. Still, though, when you do that correctly ζ(-2) = 0. But if you look at that sum,
ζ(-2) "=" 12 + 22 + 32 + 42 + ...
So does this mean that the sum of all the squares is zero? Similarly, the sum of all the fourth powers should be ζ(-4) = 0. So the sum of all the squares which are not fourth powers must also be zero, and so on ad nauseam.
From The n-Category cafe: David Corfield writes about Rota's distinction between "Algebra 1" and "Algebra 2", roughly speaking algebraic geometry and number theory versus the more combinatorial parts of algebra. My first thought here is that it's a useful distinction to make. My second is that the names are unfortunate, because when I hear "Algebra 1" and "Algebra 2" I think of classes one typically takes in high school where one learns to manipulate linear, quadratic, etc. equations.
From Math Notations: percentages are confusing to students>
From Statistical Modeling, Causal Inference, and Social Science: a map of the results of some U.S. election (I'm not sure which one, but if I had to guess I'd say 2004 presidential) with red and blue dots representing Republican and Democratic voters; each dot represents a certain number of people. I like this idea; if you remember seeing the usual political maps there's a lot more red than blue, which seems wrong because the country is evenly split (as we saw in the 2000 and 2004 presidential elections). This seems to be a nice solution. I find myself fascinated by the fact that in the middle part of the country the population centers seem to fall on a square grid. I suspect this is due to the influence of the road system consisting mostly of north-south and east-west roads, with population centers appearing at the intersections of these roads; my instinct is that a triangular lattice would be more "efficient" somehow, although I have some difficulty articulating why. Someone speculates this is the work of Robert Vanderbei, whose web site is full of pretty pictures of things mathematical.