Let $\mathrm{S}=\frac{4}{19}+\frac{44}{{19}^2}+\frac{444}{{19}^3}+\cdots \text{ up to }\mathrm{\infty }$. Then S is equal to

$\begin{array}{l}\mathrm{S}=\frac{4}{19}+\frac{44}{{19}^2}+\frac{444}{{19}^3}+\cdots \left(1\right)\\ \Rightarrow \frac{1}{19}\mathrm{S}=\frac{4}{{19}^2}+\frac{44}{{19}^3}+\cdots \left(2\right)\end{array}$

Subtracting (2) from (1), we get

$\begin{array}{c}\frac{18}{19}\mathrm{S}=\frac{4}{19}+\frac{40}{{19}^2}+\frac{400}{{19}^3}+\cdots \\ =\frac{\frac{4}{19}}{1-\frac{10}{19}}=\frac{4}{9}\\ \Rightarrow \mathrm{S}=\frac{38}{81}\end{array}$