So, this strikes me as representative of the discussion thus far.
Euler, Hardy, Ramanujan advocated exactly this. More recently, here's Terrance Tao advocating this: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/I don't see anyone advocating this.
I just came across the Cesaro sum http://en.wikipedia.org/wiki/Summation_of_Grandi's_series which Wikipedia says is "rigourous", and seems very close to what was being presented in the OP's video.That said, the youtube video cited in the original post is an example of "physicists doing math (badly)".
Is it right to say that 1+2+3+... is also not Abel summable?atyy, Cesaro summation is a rigorous method of assigning values to series. The series 1+2+3+.... is not Cesaro summable though, so it doesn't help with the original problem. It is used in some areas to great effect; for example partial sums of Fourier series can be quite bad on continuous functions even (diverging everywhere), but the Cesaro sum ends up converging for functions that are way worse than continuous. It ends up that when you want to prove results about Fourier transforms it's a lot better to consider the Cesaro sum of the Fourier series rather than the regular sum.
Notice how the wikipedia article then goes on in a later section to describe a whole host of issues in which inserting zeroes into sums can change the Cesaro sum, so when doing Cesaro summation you have to be extremely careful that you are being actually rigorous, and not just taking the word rigorous and slapping it onto a bad argument.