\(E=1-\frac{1}{2^2}-\frac{1}{3^2}-..........-\frac{1}{2004^2}\)

\(E=1-\left(\frac{1}{2^2}+\frac{1}{3^2}+..........+\frac{1}{2014^2}\right)\)

Ta có : \(E< 1-\left(\frac{1}{1.2}+\frac{1}{2.3}+..+\frac{1}{2003.2004}\right)\\ \)

Đặt A= \(1-\left(\frac{1}{1.2}+\frac{1}{2.3}+......+\frac{1}{2003.2004}\right)\\ =>A=1-\left(1-\frac{1}{2004}\right)\\ =>A=1-\frac{2003}{2004}\\ =>A=\frac{1}{2004}\)

Chắc chắn bạn đã ghi nhầm dấu 

\(\frac{1}{\left(n+1\right)\sqrt{n}}=\frac{\sqrt{n}}{\left(n+1\right)n}=\sqrt{n}\left(\frac{1}{n}-\frac{1}{n+1}\right)\)

\(=\sqrt{n}\left(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}}\right)\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)< \sqrt{n}\left(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n}}\right)\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)=2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

\(P=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+...+\frac{1}{2005\sqrt{2004}}\)

\(\Rightarrow P< 2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2004}}-\frac{1}{\sqrt{2005}}\right)\)

\(\Rightarrow P< 2\left(1-\frac{1}{\sqrt{2005}}\right)< 2.1=2\)

Lời giải:
Xét số hạng tổng quát \(\frac{1}{(n+1)\sqrt{n}}\):

\(\frac{1}{(n+1)\sqrt{n}}=\frac{(n+1)-n}{(n+1)\sqrt{n}}=\frac{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}{\sqrt{n+1}.\sqrt{n(n+1)}}\)

\(< \frac{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}{\frac{\sqrt{n+1}+\sqrt{n}}{2}.\sqrt{n(n+1)}}\)

\(\Leftrightarrow \frac{1}{(n+1)\sqrt{n}}< 2.\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n(n+1)}}=2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

Cho $n=1,2,....,2004$

\(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{2005\sqrt{2004}}< 2\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{2004}}-\frac{1}{\sqrt{2005}}\right)\)

\(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{2005\sqrt{2004}}< 2(1-\frac{1}{\sqrt{2005}})< 2\) (đpcm)

Mẫu số = 2004/1 + 2003/2 + 2002/3 + ... + 1/2004

              = (1 + 1 + ... + 1) + 2003/2 + 2002/3 + ... + 1/2004

                       2004 số 1

            = (1 + 2003/2) + (1 + 2002/3) + ... + (1 + 1/2004) + 1

            = 2005/2 + 2005/3 + ... + 2005/2004 + 2005/2005

            = 2005 × (1/2 + 1/3 + ... + 1/2004 + 1/2005)

=> B = 1/2005

Mẫu số = 2004/1 + 2003/2 + 2002/3 + ... + 1/2004

              = (1 + 1 + ... + 1) + 2003/2 + 2002/3 + ... + 1/2004

                       2004 số 1

            = (1 + 2003/2) + (1 + 2002/3) + ... + (1 + 1/2004) + 1

            = 2005/2 + 2005/3 + ... + 2005/2004 + 2005/2005

            = 2005 × (1/2 + 1/3 + ... + 1/2004 + 1/2005)

=> B = 1/2005