I have read this nice post and I have found it really interesting. The reason is the kind of approach of Lubos Motl, being physicist-like, on such somewhat old mathematical matter. The question of divergent series and their summation is as old as at least Euler and there is a wonderful book written by a great British mathematician, G. H. Hardy, that treats this problem here. Hardy is well-known for several discoveries in mathematics and one of this is Ramanujan. He had a long time collaboration with John Littlewood.
Hardy’s book is really shocking for people that do not know divergent series. In mathematics several well coded resummation techniques exist for these series. With a proper choice of one of these techniques a meaning can be attached to them. A typical example can be
and this is true exactly in the same way is true that the sum of all integers is -1/12. Of course, this means that discoveries by string theorists are surely others and most important than this one that is just good and known mathematics.
I agree with Lubos that these techniques are not routinely taught to physics students and come out as a surprise also to most mathematics students. I am convinced that Hardy’s book can be used for a very good series of lectures, for a short time, to make people acquainted with this deep matter that can have unexpected uses.
I think that mathematicians have something to teach us that is really profound: Do not throw anything out of the window. It could turn back in an unexpected way.
Update: I have three beautiful links about this matter that is very well explained leaving readers with a puzzle: