Integer S most close to A and less than A

$A=\sqrt{1^2+\dfrac{1}{1^2+2^2}}+\sqrt{1^2+\dfrac{1}{2^2+3^2}}+\sqrt{1^2+\dfrac{1}{3^2+4^2}}+—+\sqrt{1^2+\dfrac{1}{2011^2+2012^2}}$
please find an…

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