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The typical rational function has the form p(x)/q(x) where p and q are polynomials. p(x) is called the numerator and q(x) is called the denominator. Example: In the rational function. which simplifies to simplifies to (x+2)/(x-3), but the x = 2 is NOT in the domain of g(x) whereas it IS in the domain of (x+2)/(x-3).
Every function which has derivatives of all orders at 0 is the generating function of its Taylor expansion at 0. > 1/(1-x)=series(1/(1-x),x,20); Uses of generating functions. One use for generating functions is get closed formulas for sequences which are defined recursively in terms of previous terms. The defining equations for such sequences are called recurence relations.