The notion of 1+2+3+4+...=-1/12 has become so wide spread that quite a few of my friends actually feel convinced that it's actually a true statement in general, even though it is very counter intuitive.
The point is, in order to evaluate an expression, that expression has to be well-defined.
Expressions like a+b is well defined in the sense that we know it is a binary operation, we know what are we supposed to do to get the result.
Expressions like a+b+c+... is not well defined, because there are multiple operators involved, and the order of evaluation is not clearly specified.
Take the sum of the alternating series as an example: