Why do convergence issues not play a role when talking of generating functions This question has been on my mind for some while now. Perhaps it has a very simple answer.
When we talk of generating functions we take a series $\sum a_ix^i$ and do not usually bother about it as a power series? I have seen it written that usually it doesn't matter what the exact range of convergence is as long it converges for at least one non zero $x$. Why is that? Can someone explain? 

I have thought about it and I feel the answer to my question is this:
The reason for requiring convergence at a non zero point is as follows. As soon as we have convergence at a non zero $x$, by a theorem of analysis it follows that there is convergence in an open interval around $0$. Now most of the time there is a closed form expression of the analytic function for $\sum a_ix^i$, say as $f(x)$. What's important is that $f(x)$ has a unique power series expansion in that interval and so we are guaranteed that there is a one-one correspondence between the purely discrete sequence $(a_n)$ and the analytic function defined by $f(x)$. This is usually exploited in the reverse direction, for if we wish to recover our sequence from the function defined by $f(x)$ there is absolutely no ambiguity, and we cannot get back any other sequence. In fact, we may say that our sequence has been encoded within $f(x)$ as a closed form expression.
If convergence was only given at $0$, then such a one-one correspondence is not possible, since any closed form analytic function $f$, which is $a_0$ at $0$ would correspond to the sequence $(a_0,a_1,a_2,\cdots)$.
I will add that this question was regarding generating functions at an elementary level, to solve recurrence relations etc. It may be that there are viewpoints where convergence becomes of paramount importance, or of no importance at all. I would be glad if someone could inform me of such situations.
