So sánh $\dfrac{{2011}}{{2012}} + \dfrac{{2012}}{{2013}} + \dfrac{{2013}}{{2011}}$ với 3.

Ta có:

$\dfrac{{2011}}{{2012}} + \dfrac{{2012}}{{2013}} + \dfrac{{2013}}{{2011}} = \dfrac{{2012 - 1}}{{2012}} + \dfrac{{2013 - 1}}{{2013}} + \dfrac{{2011 + 1 + 1}}{{2011}}$

$= 1 - \dfrac{1}{{2012}} + 1 - \dfrac{1}{{2013}} + 1 + \dfrac{1}{{2011}} + \dfrac{1}{{2011}}$

$= 3 + \left( {\dfrac{1}{{2011}} - \dfrac{1}{{2012}}} \right) + \left( {\dfrac{1}{{2011}} - \dfrac{1}{{2013}}} \right)$

Ta thấy: vì $2011 < 2012 < 2013$ nên $\dfrac{1}{{2011}} > \dfrac{1}{{2012}} > \dfrac{1}{{2013}}$.

Suy ra: $\dfrac{1}{{2011}} - \dfrac{1}{{2012}} > 0;\,\dfrac{1}{{2011}} - \dfrac{1}{{2013}} > 0$

Do đó: $3 + \left( {\dfrac{1}{{2011}} - \dfrac{1}{{2012}}} \right) + \left( {\dfrac{1}{{2011}} - \dfrac{1}{{2013}}} \right) > 3$.

Hay $\dfrac{{2011}}{{2012}} + \dfrac{{2012}}{{2013}} + \dfrac{{2013}}{{2011}} > 3.$