I still find it hard to grasp the physical meaning of using the complex plane to go around 1 to get the answer -1

follow up question: why are we allowed to do this?

Hi Johann – this concept is challenging, and I would recommend some further reading about analytic continuation, to see how these series can get mapped onto the complex plane. Here is a good overview: https://www.mathpages.com/home/kmath649/kmath649.htm

Is this related to the idea that the sum of all the positive integers is -1/12?

Hi Andrew - yes, the sum of integers [which is the Riemann zeta function for -1, ζ(-1)] being equal to -1/12 also results from applying analytic continuation.