Question

Chứng minh: \(\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{2007^2}>\frac{1}{5}\)

Answer 1

Đặt :

\(A=\frac{1}{5^2}+\frac{1}{6^2}+.........+\frac{1}{2007^2}\)

Ta thấy :

\(\frac{1}{5^2}>\frac{1}{5.6}\)

\(\frac{1}{6^2}>\frac{1}{6.7}\)

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

\(\frac{1}{2007^2}>\frac{1}{2007.2008}\)

\(\Leftrightarrow A>\frac{1}{5.6}+\frac{1}{6.7}+........+\frac{1}{2007.2008}\)

\(\Leftrightarrow A>\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+......+\frac{1}{2007}-\frac{1}{2008}\)

\(\Leftrightarrow A>\frac{1}{5}-\frac{1}{2008}>\frac{1}{5}\)

\(\Leftrightarrow A>\frac{1}{5}\)