As far as I understood, the terms in a conditionally convergent series can be rearranged so that the sum converges to any value at all. I was reading the book 'Lattice Sums, Then and Now' by Borwein et al. In one chapter they talk about the conditionally convergent series for the Madelung constant
We have provided a unity to the concept of the Madelung constant by the use of the analytic continuation of a complex function. Thus, although the expression for the Madelung constant is conditionally convergent when summed by expanding squares (or cubes), other methods of summing will provide the same answer provided that they are ‘analytic’ in the correct sense.
So there are some methods of summing that all agree with each-other, and they all produce the same "correct" evaluation of the sum. Is there terminology to describe this "correct" value, e.g. one might call it the "principle value of a conditional sum" (but I am just making that up)