$\frac{1}{4}-\frac{5}{4.8}+\frac{5.7}{4.8.12}-\dots \dots =$

$\mathrm{S}=\frac{1}{4}-\frac{5}{2!} {\left(\frac{1}{4}\right)}^2+\frac{5.7}{3!} {\left(\frac{1}{4}\right)}^3....$

$3\mathrm{S}=3\left(\frac{1}{4}\right)-\frac{3.5}{2!}{\left(\frac{1}{4}\right)}^2+\frac{3.5.7}{3!}{\left(\frac{1}{4}\right)}^3+.....$

$1-3\mathrm{S}=1-3\left(\frac{1}{4}\right)+\frac{3.5}{2!}{\left(\frac{1}{4}\right)}^2-\frac{3.5.7}{3!}{\left(\frac{1}{4}\right)}^3....$

$$\begin{gathered} ={(1+\mathrm{x})}^{-\mathrm{p}/\mathrm{q}} \mathrm{p}=3, \mathrm{q}=2, \frac{\mathrm{x}}{\mathrm{q}}=\frac{1}{4} \\ \mathrm{x}=\frac{1}{2} \end{gathered}$$

$1-3S={\left(\frac{3}{2}\right)}^{-3/2}={\left(\frac{2}{3}\right)}^{3/2}=\frac{2\sqrt{2}}{3\sqrt{3}}$

$\mathrm{S}=\frac{1}{3} \left[1-\frac{2\sqrt{2}}{3\sqrt{3}}\right]=\left[\frac{3\sqrt{3}-2\sqrt{2}}{9\sqrt{3}}\right]$