[腦力大挑戰] Mathematical analysis BB班

The "identity" 1 + 2 + 3 + ... = -1/12 is wrong, because the summation of a divergent series does not have a value and cannot be equal to a finite number.
However, the following statement is correct:
"The Riemann-zeta-regularized summation of 1 + 2 + 3 + ... yields -1/12."

Then this function zeta(s) is analytically continued to negative values of s, where the series itself has no meaningful value any more. The result of these steps is zeta(-1) = -1/12. Mark: zeta(-1) is not equal to the original divergent series, but it follows from it through the regularization steps described here.

For a physics calculation, it is only safe to apply such Riemann-zeta regularization if the regulator s is introduced at a point in the calculation where everything is still mathematically well-defined, i.e. before arriving at a divergent series. Otherwise there are caveats such that the calculation might lead to right or wrong results, you never know in general, and manipulating divergent expressions is a dangerous thing to do.