Staggered triplets

Calculus Level 4

\[S=\dfrac { 1 }{ 1\cdot 3\cdot 5 } +\dfrac { 1 }{ 2\cdot 4\cdot 6 } +\dfrac { 1 }{ 3\cdot 5\cdot 7 } +\dfrac { 1 }{ 4\cdot 6\cdot 8 } +\dfrac { 1 }{ 5\cdot 7\cdot 9 } + \cdots\]

If \(S\) can be expressed in the form \(\frac { A }{ { B }^{ C }D }\) with \(A\), \(B\), \(C\), and \(D\) being positive prime integers, find the value of \[\dfrac { A+B+C+D }{ D }.\]


This problem is original.
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