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Calculus Level 5

\[\sum _{ n=1 }^{ \infty }{ \frac { \Gamma \left( n+\frac { 1 }{ 2 } \right) }{ { n }^{ 2 }\Gamma (n) } } =A\sqrt [ B ]{ \pi } \ln { C } \]

If the above equation holds true for positive integers \(A\), \(B\) and \(C\), where \(C\) is square-free, then find \(ABC\).

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

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