What is the sum of all natural numbers? (one to infinity)

I watched a video in which somebody writes out what he says is a proof for the statement that the sum of all natural numbers (1+2+3+4+...) is -1/12.

This is my first post, so I can't post a link. But the video is done on the Numberphile youtube channel. The title is "1+2+3+4+5"

The first part of the proof (1-1+1-1+1-1+... = 1/2) does not seem to be correct. I would think that there would simply be no answer when the value oscillates between 0 and 1. What do you think?

Also, wikipedia seems to recognize that the end result is a real thing, and the guy in the video shows a physics text book with the statement in it.

Anybody know what's going on? Thanks