1 + 2 + 3 + 4 + 5 + ... = -1/12

The idea of the first sum (S1) equaling 1/2, is hypothetical, it's the 'inbetween', the whole proof is based off of it. It's a nice idea, but mathematically, its only possible when assuming the answer to a sum that cannot be summed. Other than that, its amazing, used in string theory too.

Of course this is extremely hypothetical, but there's not much else we can do with an infinite sum than to see how we can use, simplify and apply it (eg. String theory).

It cannot be summed, so its use in string theory is beyond me. Still pretty interesting.

I have no clue why either. Still interesting though :P