The latest Numberphile video is about the sum of 1+2+3+4 all the way to infinity.
The proof is surprising (and a bit controversial, judging by the viewer comments):
I filmed this with Dr Tony Padilla from the University of Nottingham
There's a second video (currently unlisted) in which Profesor Ed Copeland proves the same thing in a different way.
This second video also has extra footage cut from the original with Tony!
I always find it fun to see how many people follow links from the main video (which as gone to all the usual subscribers) and watch the "hidden extras".