I have a rather general question and would be happy if you could teach me how to answer it :-)
Consider for example $$f(x)=(1-x)^{1/3}$$ which one could represent as a power series: $$(1-x)^{1/3}=\sum_{n=0}^{\infty}\frac{\Gamma(n-1/3)}{\Gamma(n+1)\Gamma(-1/3)}x^n.$$ For $g(x)=\frac1{f(x)}$ we have $$(1-x)^{-1/3}=\sum_{n=0}^{\infty}\frac{\Gamma(n+1/3)}{\Gamma(n+1)\Gamma(1/3)}x^n.$$
Now say we define \begin{align} F_N&=\sum_0^N\frac{\Gamma(n-1/3)}{\Gamma(n+1)\Gamma(-1/3)}\\ G_N&=\sum_0^N\frac{\Gamma(n+1/3)}{\Gamma(n+1)\Gamma(1/3)}. \end{align} Clearly $F_N\to0$ and $G_N\to \infty$. And one could also show that $G_NF_N \to \frac{3\sqrt3}{2\pi}\neq 1$.
Question How can I in general analyze $F_NG_N$ when $F_N$ and $G_N$ are partial sums of the coefficients of the power series of two functions $f(x)$ and $g(x)$ where $f(x)=\frac1{g(x)}$ and that $g(x)$ diverges when $x\to 1$?
I appreciate any help and/or reference :-)