Ta có:
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+.....+\frac{1}{2014^2}\)
\(< \frac{1}{4}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+.....+\frac{1}{2013\cdot2014}\)
\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....+\frac{1}{2013}-\frac{1}{2014}\)
\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{2014}\)
\(=\frac{3}{4}-\frac{1}{2014}\)
\(< \frac{3}{4}\)