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Calculus Level 5

\[\large \lim _{ n\to \infty }{ (f(n))^{ 1/2n^{ 2 } } } \]

Let \(f(n)\) denote the number of perfect matchings which cover the \(2n\times 2n\) square planar lattice. If the limit above can be expressed in the form \(A{ e }^{ BG/\pi }\) for positive integers \(A\) and \(B\). Find the product \(AB\).

Notation: \(\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \) is the Catalan's constant.

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