Staggered triplets

Calculus Level 4

S=1135+1246+1357+1468+1579+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 SS can be expressed in the form ABCD\frac { A }{ { B }^{ C }D } with AA, BB, CC, and DD being positive prime integers, find the value of A+B+C+DD.\dfrac { A+B+C+D }{ D }.


This problem is original.
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