S=\(\dfrac{1}{1\sqrt{2}+2\sqrt{2}}+\dfrac{1}{2\sqrt{3}+3\sqrt{2}}+\dfrac{1}{3\sqrt{4}+4\sqrt{3}}+....+\dfrac{1}{99\sqrt{100}+100\sqrt{99}}\)
Rút gọn
A = \(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+...+\dfrac{1}{100\sqrt{99}+99\sqrt{100}}\)
Cho \(A=\dfrac{\sqrt{2}-\sqrt{1}}{1+2}+\dfrac{\sqrt{3}-\sqrt{2}}{2+3}+...+\dfrac{\sqrt{100}-\sqrt{99}}{99+100}\). CMR \(A< \dfrac{1}{2}\)
\(\dfrac{\sqrt{2}-\sqrt{1}}{2+1}+\dfrac{\sqrt{3}-\sqrt{2}}{3+2}+....+\dfrac{\sqrt{100}-\sqrt{99}}{100+99}\) <\(\dfrac{9}{20}\)
Chứng minh: 17 < \(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+...+\dfrac{1}{\sqrt{100}}\) < 18
\(\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+...+\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}\)
Chứng Minh
\(\dfrac{1+\dfrac{\sqrt{3}}{2}}{1+\sqrt{1+\dfrac{\sqrt{3}}{2}}}\) + \(\dfrac{1-\dfrac{\sqrt{3}}{2}}{1-\sqrt{1-\dfrac{\sqrt{3}}{2}}}\) = 1
So sánh :
\(\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{4}}+....+\dfrac{1}{\sqrt{100}}\) và 10 .
Tính giá trị biểu thức :
\(P=\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^3}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+...+\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}\)
