If,

\[\sum _{ i=1 }^{ 200 }{ \sqrt { \left( 1+\frac { 1 }{ { i }^{ 2 } } +\frac { 1 }{ { \left( 1+i \right) }^{ 2 } } \right) } } =\frac { A }{ B } \]

Find: \(A+B\) where \(A\) and \(B\) are coprime positive integers.

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