What is the limit of a special sum at the point (1,1)?

[Ackbach]

Here is this week's POTW:

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

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[Ackbach]

Re: Problem Of The Week # 256 - Mar 18, 2017

This was Problem B-3 in the 1999 William Lowell Putnam Mathematical Competition.

No one answered this week's POTW. The solution, attributed to Kiran Kedlaya and his associates, follows:

Spoiler

We first note that
\[
\sum_{m,n > 0} x^m y^n = \frac{xy}{(1-x)(1-y)}.
\]
Subtracting $S$ from this gives two sums, one of which is
\[
\sum_{m \geq 2n+1} x^m y^n = \sum_n y^n \frac{x^{2n+1}}{1-x}
= \frac{x^3y}{(1-x)(1-x^2y)}
\]
and the other of which sums to $xy^3/[(1-y)(1-xy^2)]$. Therefore
\begin{align*}
S(x,y) &= \frac{xy}{(1-x)(1-y)} - \frac{x^3y}{(1-x)(1-x^2y)} \\
&\qquad - \frac{xy^3}{(1-y)(1-xy^2)} \\
&= \frac{xy(1+x+y+xy-x^2y^2)}{(1-x^2y)(1-xy^2)}
\end{align*}
and the desired limit is
\[
\lim_{(x,y) \to (1,1)} xy(1+x+y+xy-x^2y^2) = 3.
\]