Why 1+2+3+... can be assigned -1/12

HN uses it as the standard reference for the -1/12 meme, with commenters split between enjoying it and calling it a cheat.

A Numberphile video in which physicists Tony Padilla and Ed Copeland show a seemingly astounding result that 1+2+3+4+... equals -1/12. The argument is informal and has drawn criticism for glossing over what the result actually means.