[mr_mitm]

In an introductory course to String Theory they tried to tell me that 1+2+3+4+... = -1/12.

There is some weird appeal to the Zeta function which implies this result and apparently even has some use in String Theory, but I cannot say I was ever convinced. I then dropped the class. (Not the only thing that I couldn't wrap my head around, though.)

[thaumasiotes]

The result isn't owed to the zeta function. For example, Ramanujan derived it by relating the series to the product of two infinite polynomials, (1 - x + x² - x³ + ...) × (1 - x + x² - x³ + ...). (Ok, it's the square of one infinite polynomial.)

Do that multiplication and you'll find the result is (1 - 2x + 3x² - 4x⁴ + ...). So the sum of the sequence of coefficients {1, -2, 3, -4, ...} is taken to be the square of the sum of the sequence {1, -1, 1, -1, ...} (because the polynomial associated with the first sequence is the square of the polynomial associated with the second sequence), and the sum of the all-positive sequence {1, 2, 3, 4, ...} is calculated by a simpler algebraic relationship to the half-negative sequence {1, -2, 3, -4, ...}.

The zeta function is just a piece of evidence that the derivation of the value is correct in a sense - at the point where the zeta function would be defined by the infinite sum 1 + 2 + 3 + ..., to the extent that it is possible to assign a value to the zeta function at that point, the value must be -1/12.

https://www.youtube.com/watch?v=jcKRGpMiVTw is a youtube video (Mathologer) which goes over this material fairly carefully.