What's the intuition behind the equation 1+2+3+⋯=−1121+2+3+⋯=−112 ?

The sum clearly diverges. This is indisputable. The point of the claim above, that



is to demonstrate that a sum of infinitely many terms can be manipulated in a variety of ways to end up with a contradictory result. It's an artifact of trying to do computations with an infinite number of terms.

The mathematician Srinivasa Ramanujan famously demonstrated the above as follows: Suppose the series converges to some constant, call it . Then



Now, recall the geometric power series



which holds for any . It has derivative



Taking , we end up with



and so



But as mentioned above, neither power series converges unless . What Ramanujan did was to consider the sum as a limit of the power series evaluated at :



then arrived at the conclusion that .

But again, let's emphasize that this result is patently wrong, and only serves to demonstrate that one can't manipulate a sum of infinitely many terms like one would a sum of a finite number of terms.