$\dfrac{1}{2^2}+\dfrac{1}{4^2}+....+\dfrac{1}{(2n)^2}$
$= \dfrac{1}{2^2}.\bigg(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+....+\dfrac{2}{n^2}\bigg)$
$ < \dfrac{1}{2^2}. \bigg( 1+ \dfrac{1}{1.2}+\dfrac{1}{2.3}+....+\dfrac{1}{(n-1).n}\bigg)$
$ = \dfrac{1}{2^2}.(2- \dfrac{1}{n} ) < \dfrac{1}{2}$