Trignometric limit

Calculus Level 3

limnr=12n11tan2(rπ2n)4n \displaystyle \lim _{ n\rightarrow \infty }{ \frac { \displaystyle \sum _{ r=1 }^{ { 2 }^{ n-1 }-1 }{ \tan ^{ 2 }{ \left (\frac { r\pi }{ { 2 }^{ n } } \right ) } } }{ { 4 }^{ n } } }

For positive integer nn, the limit evaluates to ab \dfrac a b for coprime positive integers a,ba,b. What is the value of a+ba+b?

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