Reading a couple of the linked posts, I thought this was the best take on it from mathematician Jordan Ellenberg (it certainly made sense to my comp-sci brain):

Reading a couple of the linked posts, I thought this was the best take on it from mathematician Jordan Ellenberg (it certainly made sense to my comp-sci brain): 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.