Find the sum of infinity of the series
$$\frac{1}{5} + \frac{3}{5^2} + \frac{1}{5^3} + \frac{3}{5^4} + .....$$

First, we shall consider the series as a summation if two infinite series: 

S1: $$\ \ \frac{\ 1}{5}\ +\ \ \frac{\ 1}{25}+\ ......$$

S2: $$\ \ \frac{\ 3}{25}\ +\ \ \frac{\ 3}{625}+\ ......$$

Taking S1: The sum of series shall be $$\ \ \frac{\ a}{1-r}\ $$, hence $$\ \ \frac{\ \ \frac{\ 1}{5}}{1-\ \frac{\ 1}{25}}\ $$, which is $$\ \frac{\ 25}{120}$$

Taking S2: The sum of series shall be $$\ \ \frac{\ a}{1-r}\ $$, hence $$\ \frac{\ \ \frac{\ 3}{25}}{1-\ \frac{\ 1}{25}}$$, which is $$\ \frac{\ 1}{8}$$

Adding S1 and S2, we get the answer as $$\ \frac{\ 1}{3}$$