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 $A + B$.

Your answer seems reasonable.
Find out if you're right!