I just saw the proof that $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts.
I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, they said that this answer shows itself in quantum physics (e.g. it has a role in why there are 26 dimensions in string theory).
Sadly though, I am no expert in quantum physics and I'm wondering are there any other physical and/or mathematical situations that this counter intuitive result shows itself?
P.S. Needless to say, it would be great if your answer doesn't need a very deep background in physics or math.
Answer
The identity as you've stated it isn't quite correct. We usually define an infinite sum by taking the limit of the partial sums. So
$$1+2+3+4+5+\dots $$
would be what we get as the limit of the partial sums
$$1$$
$$1+2$$
$$1+2+3$$
and so on. Now, it is clear that these partial sums grow without bound, so traditionally we say that the sum either doesn't exist or is infinite.
So, to make the claim in your question title, you must adopt a nontraditional method of summation. There are many such methods available, but the one used in this case is Zeta function regularization. That page might be too advanced, but it is good to at least know the name of method under discussion.
You ask why this nontraditional approach to summation might be useful in physics. The answer is that sometimes this approach gives the physically correct result. A simple example is the Casimir effect. Suppose we place two metal plates a very short distance apart (in a vacuum, with no gravity, and so on -- we assume idealized conditions). Classical physics predicts they will just be still. However, there is actually a small attractive force between them. This can be explained using quantum physics, and calculation of the magnitude of the force uses the sum you discuss, summed using zeta function regularization.
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