Can you generalize this?

Algebra Level 5

\[ \dfrac1{1\cdot3} + \dfrac2{1\cdot3\cdot5} + \dfrac3{1\cdot3\cdot5\cdot7} + \cdots \]

If the sum above can be expressed as \(\dfrac AB\), where \(B\) is a prime and \(A\) is a positive integer, find \((A+B)!\).

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