According to the wikipedia, the method you use in this video is the Cesaro Summation. For the Grandhi series (1, - 1, 1, -1 ... ) it makes perfect sense to use this method. However, also according to the Wikipedia article, you can't do the same with a series that is divergent to positive or negative infinity, this method can't be used.
"Thus, this sequence diverges to infinity as well as G, and G is now not Cesàro summable. In fact, any series which diverges to (positive or negative) infinity the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable."
Your colleague video about the MinutePhysics Math trolling took actually the same issue: MinutePhysics Maths Trolling (Adding Past Infinity)