What kind of a sequence is this?

$S = 1 + \frac{8}{19} + \frac{88}{19^2} + \frac{888}{19^3}+ ...$

$\frac{S}{19} = \frac{1}{19}+\frac{8}{19^2}+\frac{88}{19^3} + ...$

This implies that you have:

$\frac{10S}{19} = \frac{10}{19}+\frac{80}{19^2}+\frac{880}{19^3} + ... = \frac{10}{19} + (\frac{88}{19^2} + \frac{888}{19^3} + ...) - (\frac{8}{19^2}+\frac{8}{19^3} + ...)$

$\frac{10S}{19} = \frac{10}{19} + (S - \frac{27}{19}) - \frac{4}{171} \implies S = \frac{157}{81} \approx \boxed{1.938}$

$S = 1 + \frac{\frac{8}{9}(10^1-1)}{19^1} + \frac{\frac{8}{9}(10^2-1)}{19^2} + ...$

$= 1 + \frac{8}{9}\sum\limits_{n=0}^\infty \frac{10^n-1}{19^n}$

$=1 + \frac{8}{9}\sum\limits_{n=0}^\infty \left[ \frac{10^n}{19^n} - \frac{1}{19^n} \right]$

$=1 + \frac{8}{9}\left[ \frac{1}{1-\frac{10}{19}} - \frac{1}{1-\frac{1}{19}} \right]$

$S=\frac{157}{81} \approx \boxed{1.938}$