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The sum of all positive integers

What do you think you get if you add 1+2+3+4+5+… all the way on up to infinity? Probably a massively huge number, right? Nope. You get a small negative number:

This is, by a wide margin, the most noodle-bending counterintuitive thing I have ever seen. Mathematician Leonard Euler actually proved this result in 1735, but the result was only made rigorous later and now physicists have been seeing this result actually show up in nature. Amazing. (thx, chris)

Update: Of course (of course!) the actual truth seems more complicated, hinging on what “sum” means mathematically, etc. (via @cenedella)

Update: As usual, Phil Plait sorts things out on this complicated situation. (via @theory)