## Regularization

I don't think I have even the general idea of regularization down straight. I have two possible ways to interpret it, and I doubt if either one is correct. I would be grateful for correction:

First way: given a infinite sum S , construct a related sum F(p) with parameter p which goes to S(p) as p approaches a certain value, but which converges at that value, and take this value.

Second way: given an infinite sum S which diverges in the values that we are interested in, find an analytic continuation of S which converges in that value, and use that.

Furthermore, in both ways I am not sure what allows us to use the finite value instead of the infinite one. The famous story of Euler summing a series this way and getting -1/12 by this method, as opposed to infinity in the other method, and him writing something sarcastic like "-1/12 = infinity. Great is the glory of God" in his notes illustrates my confusion.