Simplify It

Algebra Level 4

\[ \sqrt{1+\frac{1}{1^2} + \frac{1}{2^2} } + \sqrt{1+\frac{1}{2^2} + \frac{1}{3^2} } + \cdots + \sqrt{1+\frac{1}{2015^2} + \frac{1}{2016^2} } = \, ? \]

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