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Calculus Level 5

\[ \large \displaystyle\sum _{ k=1 }^{ \infty }{ \dfrac { \zeta (k+1)-1 }{ \binom{2+k}{k} } } =A\ln { (B\pi ) } -C\gamma -D \]

The above equation holds true for positive integers \(A,B,C\) and \(D\). Find \(A+B+C+D\).

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