1999 Putnam B3

Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let

$$S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,$$

where the sum ranges over all pairs $(m,n)$ of positive integers

satisfying the indicated inequalities. Evaluate

$$\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).$$

设$A=\{(x,y):0\leq x,y<1\}$。对于 A$中的$(x,y)\，令

$$S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,$$

其中总和范围涵盖所有正整数对 $(m,n)$

满足指定的不等式。评价

$$\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y)。$$