Ta có :\(\frac{2012}{\sqrt{2013}}+\frac{2013}{\sqrt{2012}}=\frac{2013-1}{\sqrt{2013}}+\frac{2012+1}{\sqrt{2012}}\)

=>\(\frac{2013}{\sqrt{2013}}-\frac{1}{\sqrt{2013}}+\frac{2012}{\sqrt{2012}}+\frac{1}{\sqrt{2012}}\)

=>\(\sqrt{2013}-\frac{1}{\sqrt{2013}}+\sqrt{2012}+\frac{1}{\sqrt{2012}}\)

Mà \(\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}>0\)

Vậy \(\sqrt{2012}+\sqrt{2013}+\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}>\sqrt{2012}+\sqrt{2013}\)

Hay \(\frac{2012}{\sqrt{2013}}+\frac{2013}{\sqrt{2012}}>\sqrt{2012}+\sqrt{2013}\)

= \(\frac{1}{\sqrt{2}\left(\sqrt{2}+1\right)}+\frac{1}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{1}{\sqrt{2012}.\sqrt{2013}\left(\sqrt{2013}+\sqrt{2012}\right)}\)

= \(\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}{\sqrt{2\left(\sqrt{2}+1\right)}}+...+\frac{\left(\sqrt{2013}-\sqrt{2012}\right)\left(\sqrt{2013}+\sqrt{2012}\right)}{\sqrt{2012}\sqrt{2013}\left(\sqrt{2012}+\sqrt{2013}\right)}\)

= \(\frac{\sqrt{2}-1}{\sqrt{2}}+...+\frac{\sqrt{2013}-\sqrt{2012}}{\sqrt{2012}\sqrt{2013}}\)

= \(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\)

= \(\frac{\sqrt{2013}-1}{\sqrt{2013}}=\frac{2013-\sqrt{2013}}{2013}\)

Xét dạng tổng quát: \(\frac{1}{\left(n+1\right)\sqrt{n}}=\frac{\sqrt{n}}{n\left(n+1\right)}=\sqrt{n}.\left(\frac{1}{n}-\frac{1}{n+1}\right)=\sqrt{n}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}}\right)\)

\(=\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(\sqrt{n}.\frac{1}{\sqrt{n}}+\sqrt{n}.\frac{1}{\sqrt{n+1}}\right)\)

\(=\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(1+\frac{\sqrt{n}}{\sqrt{n+1}}\right)< \left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(1+\frac{\sqrt{n+1}}{\sqrt{n+1}}\right)=2.\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

Thay vào đề bài ta có:

\(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2013\sqrt{2012}}\)

\(< 2.\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\right)\)

\(< 2.\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2013}}\right)\)

\(< 2.\left(1-\frac{1}{\sqrt{2013}}\right)< 2\left(đpcm\right)\)

Liên hợp

Xét dạng tổng quát :\(\frac{1}{\left(n+1\right)\sqrt{n}}=\frac{\sqrt{n}}{n\left(n+1\right)}=\sqrt{n}.\frac{1}{n\left(n+1\right)}=\sqrt{n}\left(\frac{1}{n}-\frac{1}{n+1}\right)=\sqrt{n}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}}\right)\)

\(=\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(1+\frac{\sqrt{n}}{\sqrt{n+1}}\right)< \left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\left(1+\frac{\sqrt{n+1}}{\sqrt{n+1}}\right)=2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

Áp dụng vào bài toán:

\(\frac{1}{2\sqrt{1}}< 2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\right);\frac{1}{3\sqrt{2}}< 2\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right);...;\frac{1}{2013\sqrt{2012}}< 2\left(\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\right)\)

=>\(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}...+\frac{1}{2013\sqrt{2012}}< \)\(2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\right)+2\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+2\left(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}\right)+...+2\left(\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\right)\)

mà \(2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\right)+2\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+2\left(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}\right)+...+2\left(\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\right)\)

=\(2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\right)\)

\(=2\left(1-\frac{1}{\sqrt{2013}}\right)=2-\frac{2}{\sqrt{2013}}< 2\)

Vậy \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}...+\frac{1}{2013\sqrt{2012}}< 2\)