\[\large{\lim_{n \to \infty} \left( \left( \sum_{i=1}^{2n} \dfrac{(-1)^{i-1}}{i} {2n \choose i} \right) - \ln(8n) \right)}\]

If the limit above can be expressed as \(\gamma - \ln(A)\) for positive integer \(A\), and where \(\gamma\) is the Euler-Mascheroni's Constant, find \(A\).

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