The Most Counter-Intuitive Result In Mathematics

[Donya]

I was referring to the video. There's no averaging involved from what I can tell. They just didn't go into detail why the first sum equals 1/2, which I gave of an example above.

 
The approach described on the page I linked is the same thing as in the video and also discussed in the supplementary video. The other approach you mentioned is also described here...
http/en.wikipedia.org/wiki/Grandi%27s_series#cite_note-Devlin77-1
...with the point that it does "not consider what the sum of a series actually means." In other words, it is not formal.