Bài giải:

S = $\frac{5}{27}+\frac{5}{27\times 2}+\frac{1}{6\times 3}+\frac{1}{3\times 9}+\frac{5}{3\times 63}+\frac{5}{63\times 4}+\frac{5}{4\times 81}+\frac{1}{81}$

S = $\frac{5}{27}+\frac{5}{54}+\frac{1}{18}+\frac{1}{27}+\frac{5}{189}+\frac{5}{252}+\frac{5}{324}+\frac{1}{81}$

S = $\frac{5}{27}+\frac{5}{54}+\frac{5}{90}+\frac{5}{135}+\frac{5}{189}+\frac{5}{252}+\frac{5}{324}+\frac{5}{405}$

S = $\frac{5}{9\times 3}+\frac{5}{9\times 6}+\frac{5}{9\times 10}+\frac{5}{9\times 15}+\frac{5}{9\times 21}+\frac{5}{9\times 28}+\frac{5}{9\times 36}+\frac{5}{9\times 45}$

S = $\frac{10}{9\times 6}+\frac{10}{9\times 12}+\frac{10}{9\times 20}+\frac{10}{9\times 30}+\frac{10}{9\times 42}+\frac{10}{9\times 56}+\frac{10}{9\times 72}+\frac{10}{9\times 90}$

S = $\frac{10}{9}\times (\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90})$

S = $\frac{10}{9}\times (\frac{1}{2\times 3}+\frac{1}{3\times 4}+\frac{1}{4\times 5}+\frac{1}{5\times 6}+\frac{1}{6\times 7}+\frac{1}{7\times 8}+\frac{1}{8\times 9}+\frac{1}{9\times 10})$

S = $\frac{10}{9}\times (\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10})$

S = $\frac{10}{9}\times (\frac{1}{2}-\frac{1}{10})=\frac{10}{9}\times \frac{4}{10}$

S = 4/9