Halves

Calculus Level 4

$\dfrac { 1 }{ 2 } \sqrt { \dfrac { 1 }{ 2 } +\dfrac { 1 }{ 2 } \left (\dfrac { 1 }{ 2 }\right ) } \sqrt { \dfrac { 1 }{ 2 } +\dfrac { 1 }{ 2 } \sqrt { \dfrac { 1 }{ 2 } +\dfrac { 1 }{ 2 } \left (\dfrac { 1 }{ 2 } \right) } } \cdots =\dfrac { a\sqrt { b } }{ c\pi }$

The equation above holds true for positive integers $$a,b$$ and $$c$$ with $$a,c$$ coprime and $$b$$ square-free. Find $$a+b+c$$.

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