LAST UPDATED FEBRUARY 3
Dr Tony Padilla has written his own response - read it here.
And here is a New York Times article on the video.
(And see Terry Tao's blog on this topic and analytic continuation, etc)
This is a general response to people who have messaged and emailed about our 1+2+3 video… Trying to cover a few general points rather than reply to everyone individually!
1. Firstly, thank you for taking the time to message and (in most cases) being so polite and constructive. It is great that the Numberphile audience is so passionate.
2. I think many people have not watched the second, more detailed video. I would suggest it before firing off a lengthy email.
3. For those taking issue with 1-1+1-1 etc, I would also suggest watching our video on Grandi's Series.
4. The concepts in these videos include some more obscure techniques, such as Analytical Continuation and Ramanujan Summation. It is worth reading about them.
5. There is a Wikipedia article on 1+2+3+4. Check it out.
6. Numberphile is not mathematics text book. In a short video for a popular audience you should not expect the level of detail and background explanation possible in a tome that is hundreds of pages long. Also some topics covered in the videos may involve numbers in our lives without necessarily being mathematical!
7. These are unscripted interviews. There's little point getting upset about imprecise or layman's language… I would hate to change that informal and spontaneous format. I know some of you have a PhD is mathematics and know this stuff back-to-front… But spare a thought for those who do not.
8. We often look at some advanced, obscure and controversial topics. There's no denying that some physicists do use this -1/12 result, whatever you may think of it.
DR TONY PADILLA (featured in the videos) has written a further article - here is a link.
ALSO RECOMMEND TERRY TAO'S BLOG ON THIS VERY TOPIC (and he's a Field's Medallist!)
9. On the topic of Divergent Series, the Wikipedia article is very good and I'm also advised that Chapter XIII of Konrad Knopp's book "Theory and Application of Infinite Sequences and Series" is excellent if you can find it.
10. You should NEVER disregard what your teachers or professors tell you (especially if they are setting yours tests and exams) - but it can be fun hearing what professional people are doing and the techniques they use?
11. Apparently this play - A Disappearing Number - is partly based on the result. I'm told it's quite good.
Most importantly just remember, it is a YouTube video. It is supposed to be fun and thought-provoking. Don't get upset.
Many people send very long and detailed emails… I really appreciate this but please bear in mind that a) I cannot answer them all and b) I am not a professional mathematician (though many of the interview subjects are).
A great place to leave your comments, ideas and arguments is in the YouTube comments where other people will see them, and maybe share their own views.