\[\begin{array}{l}\frac{1}{{1 + \sqrt 2 }} = \frac{{1 - \sqrt 2 }}{{\left( {1 + \sqrt 2 } \right)\left( {1 - \sqrt 2 } \right)}} = \frac{{1 - \sqrt 2 }}{{1 - 2}} = \frac{{1 - \sqrt 2 }}{{ - 1}} = \sqrt 2 - 1\\\frac{1}{{\sqrt 2 + \sqrt 3 }} = \frac{{\sqrt 2 - \sqrt 3 }}{{\left( {\sqrt 2 + \sqrt 3 } \right)\left( {\sqrt 2 - \sqrt 3 } \right)}} = \frac{{\sqrt 2 - \sqrt 3 }}{{2 - 3}} = \sqrt 3 - \sqrt 2 \\...\\\frac{1}{{1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + ... + \frac{1}{{\sqrt {2015} + \sqrt {2016} }} = \\ = \left( {\sqrt 2 - 1} \right) + \left( {\sqrt 3 - \sqrt 2 } \right) + ... + \left( {\sqrt {2016} - \sqrt {2015} } \right) = \sqrt {2016} - 1\end{array}\]

\[\sqrt {2016} - 1\]