Ta có:
\(\dfrac{1}{\sqrt{n}}=\dfrac{2}{2\sqrt{n}}\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{1}{\sqrt{n}}< \dfrac{2}{\sqrt{n-1}+\sqrt{n}}=2\left(\sqrt{n}-\sqrt{n-1}\right)\\\dfrac{1}{\sqrt{n}}>\dfrac{2}{\sqrt{n}+\sqrt{n+1}}=2\left(\sqrt{n+1}-\sqrt{n}\right)\end{matrix}\right.\)
Thế vô giải tiếp
CMR: \(\dfrac{1}{2}+\dfrac{1}{3\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+....+\dfrac{1}{2005\sqrt{2004}}< 2\)
Chứng minh : \(\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+...+\dfrac{1}{2005\sqrt{2004}}< 2\)
Rút gọn:
a) \(A=\dfrac{1}{\sqrt{3}+\sqrt{5}}+\dfrac{1}{\sqrt{5}+\sqrt{7}}+\dfrac{1}{\sqrt{7}+\sqrt{9}}+... +\dfrac{1}{\sqrt{97}+\sqrt{99}}\)
b) \(B=\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+...+\dfrac{1}{2006\sqrt{2005}+2005\sqrt{2006}}+\dfrac{1}{2007\sqrt{2006}+2006\sqrt{2007}}\)
Cho 3 số dương x,y,z. CMR:\(\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}>=3\left(\dfrac{1}{\sqrt{x}+2\sqrt{y}}+\dfrac{1}{\sqrt{y}+2\sqrt{z}}+\dfrac{1}{\sqrt{z}+2\sqrt{x}}\right)\)
\(\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+....+\dfrac{1}{\sqrt{2005}+\sqrt{2006}}\)
Giải chi tiết dùm mình hem
CMR : \(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+...+\dfrac{1}{\sqrt{100}}< 18\)
CMR :
\(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{100}}>18\)
CMR:
\(\dfrac{1}{1+\sqrt{2}}\)+\(\dfrac{1}{\sqrt{3}+\sqrt{4}}+\dfrac{1}{\sqrt{5}+\sqrt{6}}+....+\dfrac{1}{\sqrt{79}+\sqrt{80}}>4\)
Cho \(A=\dfrac{\sqrt{2}-\sqrt{1}}{1+2}+\dfrac{\sqrt{3}-\sqrt{2}}{2+3}+...+\dfrac{\sqrt{100}-\sqrt{99}}{99+100}\). CMR \(A< \dfrac{1}{2}\)
