If a generating function does not condense to some small symbolic expression, it is next to useless. Yes, the radius of convergence is not very relevant for G.F

If a generating function does not condense to some small symbolic expression, it is next to useless. Yes, the radius of convergence is not very relevant for G.F., however, without any convergence, there will be no shorthand symbolic expression. And that's really what we are after with G.F. For example 1/(1-x) for the OGF of the sequence 1,1,1,1.....etc. We don't care that the interval of convergence is between (-1,1). However if there was no convergence at all, the expression 1+x+x^2+x^3..... would not simplify.