**1/5 + 1/7 + 1/5**^{2} + 1/7^{2} + ....

= (1/5 + 1/5^{2} + 1/5^{3} + ...) + (1/7 + 1/7^{2} + 1/7^{3} + ...)

\(=\cfrac{\frac{1}{5}}{1-\frac{1}{5}}+\cfrac{\frac{1}{7}}{1-\frac{1}{7}}\) (∵ Sum of infinite series in G.P. when r < 1 is a/1-r)

\(=\cfrac{\frac{1}{5}}{\frac{4}{5}}+\cfrac{\frac{1}{7}}{\frac{6}{7}}\)

**= 1/4 + 1/6 \(=\frac{3+2}{12}\) = 5/12.**