\[\displaystyle{{ D }_{ n }\left( g \right) =\sum _{ k=0 }^{ n }{ \cfrac { { 2 }^{ k }{ g }^{ { 2 }^{ k } } }{ g(1+{ g }^{ { 2 }^{ k } }) } } }\]

Define above summation such that \(\displaystyle{\left| g \right| <1}\) .

\(\displaystyle{\lim _{ n\rightarrow \infty }{ { D }_{ n }\left( \cfrac { 1 }{ 2015 } \right) } =\cfrac { a }{ b } }\)

Find \(2a-b\)

**Details and assumptions**

\(\bullet \) Here \(a,b\) are co-primes.

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