This is making the rounds and I think this is the first time I've seen this. Apparently, the infinite sum 1 + 2 + 3 + 4 + ⋯ is -1/12. Really. And this value is useful in physics (well string theory and some other things). And I feel good because even the Bad Astronomer can't quite wrap his head around it.
This is the video going around...
After that one bothers you goto the followup page they posted with two more videos with some more details, Thanks for the messages. Each had a few bits that helped me.
There are some comments on Quora of people trying to help explain this non-obvious result. This one was useful.