Not Geometric, Not Arithmetic...

Algebra Level 4

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

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