\(S=\frac{1}{3\sqrt{1}+3\sqrt{3}}+\frac{1}{3\sqrt{3}+3\sqrt{5}}+...+\frac{1}{3\sqrt{2017}+3\sqrt{2019}}\)
\(S=\frac{1}{3}\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+...+\frac{1}{\sqrt{2017}+\sqrt{2019}}\right)\)
\(S=\frac{1}{3}\left[\frac{\sqrt{3}-\sqrt{1}}{3-1}+\frac{\sqrt{5}-\sqrt{3}}{5-3}+...+\frac{\sqrt{2019}-\sqrt{2017}}{2019-2017}\right]\)
\(S=\frac{1}{3}\cdot\frac{\sqrt{3}-\sqrt{1}+\sqrt{5}-\sqrt{3}+...+\sqrt{2019}-\sqrt{2017}}{2}\)
\(S=\frac{\sqrt{2019}-\sqrt{1}}{6}\)
\(2S=\frac{1}{3}\left(\frac{2}{\sqrt{1}+\sqrt{3}}+\frac{2}{\sqrt{3}+\sqrt{5}}+...+\frac{2}{\sqrt{2017}+\sqrt{2019}}\right)\)
\(2S=\frac{1}{3}\left(\frac{3-1}{\sqrt{1}+\sqrt{3}}+\frac{5-3}{\sqrt{3}+\sqrt{5}}+...+\frac{2019-2017}{\sqrt{2017}+\sqrt{2019}}\right)\)
\(2S=\frac{1}{3}\left(\frac{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}{\sqrt{3}+1}+\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\sqrt{3}+\sqrt{5}}+...+\frac{\left(\sqrt{2019}-\sqrt{2017}\right)\left(\sqrt{2019}+\sqrt{2017}\right)}{\sqrt{2019}+\sqrt{2017}}\right)\)
\(2S=\frac{1}{3}\left(\sqrt{3}-1+\sqrt{5}-\sqrt{3}+...+\sqrt{2019}-\sqrt{2017}\right)\)
\(S=\frac{\sqrt{2019}-1}{6}\)
