Inspired by Sharky Kesa

\[ \large \displaystyle \prod_{n=1729}^{2017} \sum_{k=1}^{\infty} \dfrac{n}{\binom{k+n}{k}}\]

If the above expression is the form of \( \dfrac AB\), where \(A\) and \(B\) are coprime positive integers, find \(A+B\).

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