\(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{n}+\sqrt{n+1}}\ge2014\)
\(\Rightarrow\frac{1-\sqrt{2}}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}+\frac{\sqrt{2}-\sqrt{3}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+...+\frac{\sqrt{n}-\sqrt{n+1}}{\left(\sqrt{n}+\sqrt{n+1}\right)\left(\sqrt{n}-\sqrt{n+1}\right)}\)
\(=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+...+\frac{\sqrt{n}-\sqrt{n+1}}{n-\left(n+1\right)}\)
\(=\frac{1-\sqrt{2}+\sqrt{2}-\sqrt{3}+...+\sqrt{n}-\sqrt{n+1}}{-1}\)
\(=\frac{1-\sqrt{n+1}}{-1}=\sqrt{n+1}-1\ge2014\)
\(\Leftrightarrow\sqrt{n+1}\ge2015\)
\(\Leftrightarrow n+1=2015^2=4060225\)
\(V~~n=4060224\)
