$\text{ The sum of the infinite series }\frac{5}{13}+\frac{55}{{13}^2}+\frac{555}{{13}^3}+\dots ..$

$\Rightarrow \begin{array}{l}We have S=\frac{5}{13}+\frac{55}{{13}^2}\mathrm{........}\\ \Rightarrow \frac{S}{13}=\frac{5}{{13}^2}+\frac{55}{{13}^3}\mathrm{........}\\ \left(S-\frac{S}{13}\right)=\frac{5}{13}+\frac{50}{{13}^2}+\frac{500}{{13}^3}\mathrm{......}\\ \Rightarrow \frac{12S}{13}=\frac{5}{13}+\frac{50}{{13}^2}+\mathrm{.....}\\ \Rightarrow S=\frac{65}{36}=1.81\end{array}$