A calculus problem by Ben Habeahan

Calculus Level 3

\[\frac {1+1}{1^2{(1+2)}^2} +\frac {2+1}{2^2{(2+2)}^2} +\frac {3+1}{3^2{(3+2)}^2} + \ldots = \frac{A}{B}\]

Find \( A+B \) where \(A\) and \(B\) are coprime positive integers.

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