The sum of all natural numbers...

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:

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