A further point on the reference to {1+2+3+4.....}=-1/12 in the book on String Theory by Joseph Polchinski is that the mathematical formalism is based on the operator method.
To save space, equations are truncated to only include the actions of the operator on the quantum states. The quantum states are not included in the equations.
The equation 1.3.32 in the
book does not illustrate the summing of the numbers from one to infinity, but the summing of the energy levels of the quantum states which are multiples of (n+1/2), where n=1,2,3........
Steven