Question

CMR; 87/89 < 1/2căn1 + 1/3căn2 + ... +1/2011căn2010 < 88/45

Answer 1

tổng quát nhé \(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}=\frac{1}{\sqrt{\left(k+1\right)k}}>\frac{1}{\left(k+1\right)\sqrt{k}}>\frac{1}{\left(k+1\right)k}=\frac{1}{k}-\frac{1}{k+1}\)
Đặt A= \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+...+\frac{1}{2011\sqrt{2010}}\)
\(\Rightarrow1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2010}}-\frac{1}{\sqrt{2011}}>A>1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2010}-\frac{1}{2011}\)\(\Leftrightarrow1-\frac{1}{\sqrt{2011}}>A>1-\frac{1}{2011}\Leftrightarrow\frac{88}{45}>\frac{2011-\sqrt{2011}}{2011}>A>\frac{2010}{2011}>\frac{87}{89}\)

Answer 2

ban chung minh tong quat ro hon ko