So right now I'm thinking about the summation of divergent sums. What do I mean by that? Well, say I wanted to add up some numbers:
That should give me some answer, right? e.g., if N=4 then I'd have
Not so bad, right? In fact, it's not hard to show (if you are good at math) that you can find the general answer
Which works for the above example because 4*(4+1)/2=10.
Obviously, the higher you count, the bigger the answer is going to be, right? But what if I wanted to never stop... in other words, what if I wanted to count all the way to infinity. "Ha ha!" you say! Such an idea is absurd! But No! You can do it! And the answer is... -1/12. That's right - I said negative one-twelfth. I know! It sounds dumb. But mathematicians and physicists have a way of doing it so that that's what you get. It's called analytic continuation of the Riemann Zeta Function. (by the way, Wikipedia is my best friend) And they use it in all seriousness! It's a majorly important part of string theory. And before you let this turn you off of string theory, it is also the method by which one calculates a completely standard, uncontroversial, physically tested (and true) phenomenon called the Casimir Effect. And this Riemann guy was a seriously important, revered mathematician of the 19th century, and his function is apparently one of the most important objects in pure mathematics.
Anyway, this is just what I've been thinking about lately. Now you know. And knowing...