Đặt \(S=\frac{1}{18}+\frac{1}{54}+\frac{1}{108}+...+\frac{1}{990}\)
\(\Rightarrow S=\frac{1}{3.6}+\frac{1}{6.9}+\frac{1}{9.12}+...+\frac{1}{30.33}\)
\(\Rightarrow3S=3.\left(\frac{1}{3.6}+\frac{1}{6.9}+\frac{1}{9.12}+...+\frac{1}{30.33}\right)\)
\(\Rightarrow3S=\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+...+\frac{3}{30.33}\)
\(\Rightarrow3S=\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+...+\frac{1}{30}-\frac{1}{33}\)
\(\Rightarrow3S=\frac{1}{3}-\frac{1}{33}\)
\(\Rightarrow3S=\frac{10}{33}\)
\(\Rightarrow S=\frac{10}{33}:3\)
\(\Rightarrow S=\frac{10}{99}\)
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