S = $\frac{1}{4^{2}}$ + $\frac{1}{6^{2}}$ + $\frac{1}{8^{2}}$ + ... + $\frac{1}{(2n)^{2}}$
S = $\frac{1}{2^{2}}$ . ($\frac{1}{2^{2}}$ + $\frac{1}{3^{2}}$ + $\frac{1}{4^{2}}$ + ... + $\frac{1}{n^{2}}$) < $\frac{1}{4}$ . ($\frac{1}{1.2}$ + $\frac{1}{2.3}$ + $\frac{1}{3.4}$ +...+ $\frac{1}{(n-1).n}$)
⇒ S = $\frac{1}{2^{2}}$ . ($\frac{1}{2^{2}}$ + $\frac{1}{3^{2}}$ + $\frac{1}{4^{2}}$ + ... + $\frac{1}{n^{2}}$) < $\frac{1}{4}$ . (1 - $\frac{1}{2}$ + $\frac{1}{2}$ - $\frac{1}{3}$ + $\frac{1}{3}$ - $\frac{1}{4}$ +...+ $\frac{1}{n-1}$ - $\frac{1}{n}$)
⇒ S = $\frac{1}{2^{2}}$ . ($\frac{1}{2^{2}}$ + $\frac{1}{3^{2}}$ + $\frac{1}{4^{2}}$ + ... + $\frac{1}{n^{2}}$) < $\frac{1}{4}$ . (1 - $\frac{1}{n}$) < $\frac{1}{4}$
⇒ S < $\frac{1}{4}$
