>You are right, we never finish picking it, like we never finish 1/2 + 1/4 + 1/8 + 1/(2^n)..., but we know the sum is 1.
For some sequences or series, you may be able to say what the limit is. For many more, convergence may be known or unknown, the limit may be known or unknown. New results will emerge in this area. But in general, a convergent sequence or series converges to a number that cannot, ever, be "known", in the sense of specifying it by computing action (one of the ways pi is "known").