What is the sum obtained when one adds up all whole numbers (1+2+3+4...) from 1 to infinity?

You are not alone in wondering how this can make sense. The Norwegian mathematician Niels Henrik Abel, whose notion of an Abel sum plays a role here, once wrote, “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.”

In modern terms, the gist of the calculations can be interpreted as saying that the infinite sum has three separate parts: one of which blows up when you go to infinity, one of which goes to zero, and minus 1/12. The infinite term, he said, just gets thrown away.

And it works. A hundred years later, Riemann used a more advanced and rigorous method, involving imaginary as well as real numbers, to calculate the zeta function and got the same answer: minus 1/12.

The minus 1/12 answer is used in quantum physics and in string theory. There are now several proofs of this outcome including one by ramanujain. There is also a youtube video on numberfile in which a simplified explanation of this result is provided.