$$
\begin{aligned}
& \text { Let } S=1+\frac{2}{3}\left(\frac{1}{8}\right)+\frac{2 \times 5}{3 \times 6}\left(\frac{1}{8}\right)^2+\frac{2 \times 5 \times 8}{3 \times 6 \times 9}\left(\frac{1}{8}\right)^3+\ldots \\
& =1+\frac{\frac{2}{3}}{1}\left(\frac{1}{8}\right)+\frac{\left(\frac{2}{3}\right)\left(\frac{5}{3}\right)}{2 !}\left(\frac{1}{8}\right)^2+\frac{\left(\frac{2}{3}\right)\left(\frac{5}{3}\right)\left(\frac{8}{3}\right)}{3 !}\left(\frac{1}{8}\right)^3+\ldots \\
& =\left(1-\frac{1}{8}\right)^{-\frac{2}{3}}\left[\because(1-x)^{-n}=1+n x+\frac{n(n+1)}{2 !} x^2+\ldots\right] \\
& =\left(\frac{7}{8}\right)^{-\frac{2}{3}}=\left(\frac{8}{7}\right)^{\frac{2}{3}}=\left(\frac{64}{49}\right)^{\frac{1}{3}}=\frac{(64)^{\frac{1}{3}}}{\sqrt[3]{49}}=\frac{4}{\sqrt[3]{49}}
\end{aligned}
$$