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Calculus Level 5

\[\displaystyle\sum _{ n=2 }^{ \infty }{ \dfrac { \zeta (n)-1 }{ n+2 } } =\dfrac { C }{ H } -I\ln { A } -\dfrac { C_{ 1 }\gamma ^{ K } }{ P } -\dfrac { E\ln { (A_{ 1 }\pi ) } }{ S } \]

The equation holds true, where \(A\) denotes the Glaisher Kinkelin constant and \(\gamma\) denotes the Euler-Mascheroni constant.

And \(C,H,I,C_1,K,P,E,A_1,S\) are positive integers with \(\gcd(C,H)=\gcd(C_1,P)=\gcd(E,S)=1\).

Find \(C+H+I+C_1+K+P+E+A+S\).

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