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It is a well established and non-controversial result, though, first established by Ramanujan (https://en.wikipedia.org/wiki/Ramanujan_summation).
The "trick" is that you can't really say that an infinite series (symbolized by Σ(n), for n = 1 to ∞) is equal to ∞, because ∞ is not a number. So if you insist on using the equal sign, you must regulate or regularize the infinite series.
One way to do that (not the only one!) is to stop the summation at some arbitrary step N, but that's a very blunt way that doesn't get you any closer to a useful result (because you could always add N+1 and invalidate it).
Or, you can use a smoother regulator, and gradually give less and less weight to each n as you go down the number line. It's best explained in this video from the playlist: https://youtu.be/beakj767uG4
It's using this class of regulators which provably shows that the sum is, in fact, equal to -1/12.
In fact this is regularization is crucial to quantum field theory, to avoid getting infinite results 1 reply
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