That's too much Gamma

Algebra Level 3

\[ \large \sum_{n=1}^{1729} \dfrac{\Gamma(n) \times \Gamma(n+1)}{\Gamma(n+1) \times \Gamma(n+2) } \]

If the summation above is the form of \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

Notation: \( \Gamma(\cdot) \) denotes the Gamma function.

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