An algebra problem by Tarit Goswami

Algebra Level 3

\[\small \sqrt{1+\frac 1{1^2} +\frac 1{2^2}} + \sqrt{1+\frac 1{2^2} +\frac 1{3^2}} + \sqrt{1+\frac 1{3^2} + \frac 1{4^2}}+\cdots + \sqrt{1+ \frac 1{1999^2} + \frac 1{2000^2}}= \, ? \]

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