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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