\(A=\frac{1}{2^2}+\frac{1}{4^2}+...+\frac{1}{\left(2n\right)^2}\)
\(=\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\right)\)
Có : \(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(...\)
\(\frac{1}{n^2}< \frac{1}{\left(n-1\right)n}\)
\(\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{\left(n-1\right)n}\)
\(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}\)
\(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 2-\frac{1}{n}< 2\)
\(\Rightarrow A< \frac{1}{2^2}.2=\frac{1}{2}\)
