`Answer:`
\(S=\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}\)
\(S=\frac{1}{4.4}+\frac{1}{6.6}+\frac{1}{8.8}+...+\frac{1}{2n.2n}\)
\(S< \frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{\left(2n-2\right).2n}\)
\(S< \frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+...+\frac{2}{\left(2n-2\right).2n}\right)\)
\(S< \frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2n-2}-\frac{1}{2n}\right)\)
\(S< \frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n}\right)\)
\(S< \frac{1}{4}\)
