Why do individual terms go till infinity in a generating function even when a limit is specified?

Question

For example: What is the generating function for: x1+x2+x3=12; x1>=2; 3<=x<=6; x3<=9

Why does the first term go till infinity instead of x^12. (x^2 + x^3 + x^4 +....)(x^3 + x^4 + x^5 + x^6)(1 + x + x^2 + ...)

Answer 1

In working with generating functions, we usually seek the more general solution to a problem. So rather then finding the number of solutions in integers to $$x_1 + x_2 + x_3 =12$$ subject to whatever constraints you have, we would usually prefer to find the generating function of the number of solutions to $$x_1 + x_2 + x_3 =r$$ where $r$ is an unspecified integer. Once we have the solution to the more general problem, the original problem is then just the special case $r = 12$.

Why seek the more general solution? Well, it's good to have the solution to a more general problem. And often the more general problem is easier to solve.