\(A=\dfrac{1}{\sqrt{1.1999}}+\dfrac{1}{\sqrt{2.1998}}+...+\dfrac{1}{\sqrt{1999.1}}>\dfrac{1}{\dfrac{1+1999}{2}}+\dfrac{1}{\dfrac{2+1998}{2}}+...+\dfrac{1}{\dfrac{1999+1}{2}}\)
\(=\dfrac{1}{1000}+\dfrac{1}{1000}+...+\dfrac{1}{1000}=1,999\)
\(B=\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{47}}+\dfrac{1}{\sqrt{48}}\). CMR: B > 12
a+b+c=\(\sqrt{a}+\sqrt{b}+\sqrt{b}=2\)
Cmr: \(\dfrac{\sqrt{a}}{1+a}+\dfrac{\sqrt{b}}{1+b}+\dfrac{\sqrt{c}}{1+c}=\dfrac{2}{\sqrt{\left(1+a\right)\left(1+b\right)\left(1+c\right)}}\)
CMR : \(\dfrac{1}{\sqrt{1\cdot199}}+\dfrac{1}{\sqrt{2\cdot198}}+\dfrac{1}{\sqrt{3\cdot197}}+...+\dfrac{1}{\sqrt{199\cdot1}}>1,99\)
cho a,b dương và c ≠ 0 thỏa mãn \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\). CMR: \(\sqrt{a+b}=\sqrt{b+c}+\sqrt{c+a}\)
\(A=1+\left(\dfrac{2a+\sqrt{a}-1}{1-a}-\dfrac{2a\sqrt{a}-\sqrt{a}+a}{1-a\sqrt{a}}\right).\dfrac{a-\sqrt{a}}{2\sqrt{a}-1}\)
a)rút gọn
b) tìm a để A=\(\dfrac{\sqrt{6}}{1+\sqrt{6}}\)
c)CMR A>\(\dfrac{2}{3}\)
CMR : \(2\left(\sqrt{n+1}-\sqrt{n}\right)< \dfrac{1}{\sqrt{n}}< 2\left(\sqrt{n}-\sqrt{n-1}\right)\) với n thuộc N*
Áp dụng cho : \(A=1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{100}}\) . CMR : 18 < A < 19
CMR:
\(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{3}}+....+\dfrac{1}{\left(n+1\right)\left(\sqrt{n}+n\sqrt{n+1}\right)}< 1\)
CMR : \(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{2018}}>2\left(\sqrt{2018}-1\right)\)
