ta có: \(\left(\sqrt{2017^2-1}-\sqrt{2016^2-1}\right)\left(\sqrt{2017^2-1}+\sqrt{2016^2-1}\right)\)
= 20172-1 - (20162-1)
= 20172-20162
= 2017+2016 > 2.2016
=> \(\sqrt{2017^2-1}-\sqrt{2016^2-1}\)\(>\) \(\frac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)
\(\sqrt{2017^2-1}-\sqrt{2016^2-1}=\frac{2017^2-1-\left(2016^2-1\right)}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}=\)\(\frac{2017^2-1-2016^2+1}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}=\frac{2017^2-2016^2}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)\(=\frac{\left(2017-2016\right)\left(2016+2017\right)}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}=\frac{2016+2017}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)\(>\frac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)
