Question

Cho \(A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2014^2}\) . Chứng tỏ \(A< \frac{3}{4}\)

Answer 1

\(\frac{1}{2^2}=\frac{1}{2\cdot2}< \frac{1}{1.2}\)

\(\frac{1}{3^2}=\frac{1}{3\cdot3}< \frac{1}{2\cdot3}\)

.......................................

\(\frac{1}{2014^2}=\frac{1}{2014\cdot2014}< \frac{1}{2013\cdot2014}\)

\(\Rightarrow A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{2013\cdot2014}\)

\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(\Rightarrow A< 1-\frac{1}{2014}=\frac{2013}{2014}\)

Answer 2

Trần Nhật Dương    Chứng minh \(A< \frac{3}{4}\) mà :)) 

Answer 3

Ta có: \(\frac{1}{3^2}=\frac{1}{3.3}< \frac{1}{2.3}\)

\(\frac{1}{4^2}=\frac{1}{4.4}< \frac{1}{3.4}\)

      ........................

\(\frac{1}{2014^2}=\frac{1}{2014.2014}< \frac{1}{2013.2014}\)

 \(\Rightarrow A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2014^2}< \frac{1}{2^2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}\)

\(\Leftrightarrow A< \frac{1}{2^2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)

\(\Leftrightarrow A< \frac{1}{2^2}+\frac{1}{2}-\frac{1}{2014}\)

\(\Leftrightarrow A< \frac{1}{4}+\frac{1}{2}-\frac{1}{2014}\)

\(\Leftrightarrow A< \frac{3}{4}-\frac{1}{2014}< \frac{3}{4}\)

Answer 4

mk nhầm