\(B=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{\left(2n\right)^2}\)
\(2B=\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{6^2}+...+\frac{1}{\left(2n\right)^2}+\frac{1}{\left(2n\right)^2}\)
\(< \frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{\left(2n-1\right)^2}+\frac{1}{\left(2n\right)^2}\)
\(< \frac{1}{2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{\left(2n-2\right)\left(2n-1\right)}+\frac{1}{\left(2n-1\right)2n}\)
\(=\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{2n-1}-\frac{1}{2n}\)
\(=1-\frac{1}{2n}< 1\)
Suy ra \(B< \frac{1}{2}\).