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\(\frac{2014}{\sqrt{2015}}+\frac{2015}{\sqrt{2014}}=\frac{2015-1}{\sqrt{2015}}+\frac{2014+1}{\sqrt{2014}}\)
= \(\sqrt{2014}+\sqrt{2015}+\frac{1}{\sqrt{2014}}-\frac{1}{\sqrt{2015}}>\sqrt{2014}+\sqrt{2015}\)
Nhân cả 2 với (\(\sqrt{2015^2-1}\)+\(\sqrt{2014^2-1}\))
A = 2015^2 -1 -2014^2 + 1 = (2014 + 1)^2 -2014^2 = 2.2014 + 1
B = 2.2014
=> A = B + 1
Có Ta có\(VT=\frac{2014}{\sqrt{2015}}+\frac{2015}{\sqrt{2014}}=\frac{2015-1}{\sqrt{2015}}+\frac{2014+1}{\sqrt{2014}}=\sqrt{2015}-\frac{1}{\sqrt{2015}}+\sqrt{2014}+\frac{1}{\sqrt{2014}}.\)\(2014<2015\Leftrightarrow\sqrt{2014}<\sqrt{2015}\Leftrightarrow\frac{1}{\sqrt{2014}}>\frac{1}{\sqrt{2015}}\Leftrightarrow\frac{1}{\sqrt{2014}}-\frac{1}{\sqrt{2015}}>0\Leftrightarrow VT>VP\)
\(\frac{1}{1+\text{ }\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{2014}+\sqrt{2015}}\)
\(=\frac{1-\sqrt{2}}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}+\frac{\sqrt{2}-\sqrt{3}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+..+\frac{\sqrt{2014}-\sqrt{2015}}{\left(\sqrt{2014}+\sqrt{2015}\right)\left(\sqrt{2014}-\sqrt{2015}\right)}\)
\(=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+...+\frac{\sqrt{2014}-\sqrt{2015}}{2014-2015}\)
\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{2005}-\sqrt{2004}=\sqrt{2005}-1\)
dangj tổng quát : cmr :\(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n}-\sqrt{n+1}\left(\right)dùngtrụccăthứcởmẫu\left(\right)\)