Excel in math, science, and engineering

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\[\mathfrak{G}=1-\displaystyle \sum^{\infty}_{n=1} \dfrac{1}{1729^n n(n+1)} \]

If the value of \(\mathfrak{G}\) is in the form \(\ln \left(1+\frac{1}{n}\right)^n\), find \(n\).

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