A calculus problem by Ben Habeahan

Calculus Level 4

1+112(1+2)2+2+122(2+2)2+3+132(3+2)2+=AB\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 A+B where AA and BB are coprime positive integers.

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