This video, described as mind-bending or mind-melting, made the rounds last week. I almost blogged about it since the result is pretty cool, but somewhere along the line it stopped making sense to me. For me it was when the shifted the series by 1 and started adding them together. I don’t have the math background to know if that’s allowed, but it struck me as strange enough that I didn’t post the video.
Well, @BadAstronomer did post it and I’m glad he did. He has since posted a clarification of the math used. It turns out you can’t “just add” divergent series like that. So I was wrong in how the video was wrong, but I was right that it was wrong… got it?
You should read the entire explanation, but toward the end there’s a quote discussing what it means for an “infinite sum” to be equal to something. It sort of puts the whole thing in a little perspective.
It’s not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 +…. = -1/12. When we ask “what is the value of the infinite sum,” we’ve made a mistake before we even answer! Infinite sums don’t have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video. A different protocol assigns it the value -1/12. Neither answer is more correct than the other.