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Calculus Level 5

\[\large \sum _{ n=0 }^{ \infty }{ \dfrac { { 2 }^{ n+1 }n!^{ 2 } }{ (2n+1)!(n+1)^{ 2 } } } =A\pi G-\dfrac { B }{ C } \zeta (D)+\dfrac { { \pi }^{ E } }{ F } \ln { H } \]

The equation above holds true for positive integers \(A,B,C,D,E,F\) and \(H\), where \(B,C\) are coprime and \(H\) is minimized.

Find \(A+B+C+D+E+F+H\).

Notations:

  • \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).
  • \(G\) denotes the Catalan's constant, \(\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \).
  • \(\zeta(\cdot) \) denotes the Riemann zeta function.
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