Not Geometric, Not Arithmetic...

Algebra Level 4

If the infinite series S=n=02n+122n+1 { S }_{ \infty } = \sum _{ n=0 }^{ \infty }{ \frac { 2n+1 }{ { 2 }^{ 2n+1 } } } can be represented as AB \frac { A }{ B } , where AA and BB are positive coprime integers, then evaluate A+B A + B .

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