\[ \large \displaystyle\sum _{ k=1 }^{ \infty }{ \displaystyle\sum _{ n={ 2 }^{ k-1 }+1 }^{ { 2 }^{ k } }{ \frac { k }{ 2n(2n-1) } } } \]

If the value of the series above is equal to \( A-B\gamma \), where \(A\) and \(B\) are positive integers, find \(A+B\).

**Notation**: \(\gamma\) denotes the Euler-Mascheroni constant.

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