In that particular point in the article, n is both. It is a natural number that is the domain of Fib, and the iterating variable of the generating function. As you've noticed, the sum is indeed infinite - this does not matter, however, as at no point is the actual sum of a generating function series important - the whole series mechanism is just a nice conduit to reason about the coefficients (subsequent values of Fib, in this case) using more familiar and developed mathematics for infinite series (divergent or not).
I remember being fascinated by generating functions in college. I viewed them as sort of a beautiful hack in how to tie various combinatorial structures to the more ordinary algebraic operations. This mini-field is called enumerative combinatorics. The only part of discrete mathematics that I've actually enjoyed :)