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I have to disagree, and am quite bemused by the moral tone taken up by some of the detractors of this work. It's too easy to dismiss these videos as incorrect/untruthful etc. In context, they are a valid exploration of mathematical ideas, and thus important and valuable.
In the history of maths, this happened with irrational numbers, negative numbers, imaginary numbers. Formal manipulations that included "nonsense" ended up enriching mathematics. And in modern maths, look at p-adics where, for example. in the 5-adics, 5+5^2+5^3+... converges but 1/5 + 1/5^2+1/5^3+... does not. There's the Umbral Calculus where the formal basis is still only being constructed. Also, the extended complex plane where infinity is just a point like any other.
Now you can say they sneakily departed from real numbers, but the whole question naturally goes beyond the reals because it involves infinity. Infinity is not a real number, but more relevant, it is not just a "really big number". That it is qualitatively different is important to learn, and apparent from ideas like divergent and non-absolutely convergent series. As a 15 year old I was introduced to 1+2+4+8+...=-1 by a maths professor on an excursion, and it made a tremendous and positive impression on me and my fellow students. And I use similar things to both communicate my love of maths, and to encourage others to look at it differently and find their own sources of wonder.
And it is not so far removed from school maths. We teach that infinity - infinity is undefined arithmetically, but in terms of sets we often show that it can have many values (easiest example is remove all even numbers from 1,2,3,4,5... and you are left with an infinite set, compared to match up 1<->2, 2<->4, 3<->6 etc and show that none are left behind). Exposure to such conflicts is a great and fun way to learn about the limits and context of maths. And a major source of error is not learning the limits, applying things unthinkingly out of context, because too much exposure is only to "nice" examples. There is loads of education research on this. These paradoxical results force students to confront the limits, and thus can be used to enhance their mathematical thinking.
The authors of the article recognise the broader context (although were clearly not entirely happy with the presentation). The commenter above, Matt E, a maths teacher, seems to miss this broader context. Of course, approaching and crossing boundaries may well mean things need to be redefined, concepts generalised, but that's mathematics. Use it to generate interest, provide historical and real-world context, and thus enrich teaching.
Thinking about Grandi's series is like a first step on a journey. Enjoy it, and let students enjoy it too. Relate it to Thomson's Lamp. Bring mathematics to life!
p.s. Martin Gardner is great, certainly, but for a bit less puzzle orientation I recommend "The Heart of Mathematics: An Invitation to Effective Thinking", by Edward Burger and Michael Starbird.