`Answer:`
Đặt \(N=\frac{9}{10}+\frac{9}{10^2}+\frac{9}{10^3}+...+\frac{9}{10^{10}}\)
\(\Rightarrow\frac{N}{9}=\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+...+\frac{1}{10^{10}}\)
Ta có \(\frac{N}{90}=\frac{1}{10^2}+\frac{1}{10^3}+\frac{1}{10^4}+...+\frac{1}{10^{11}}\)
\(\Rightarrow\frac{N}{9}-\frac{N}{90}=\left(\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+...+\frac{1}{10^{10}}\right)-\left(\frac{1}{10^2}+\frac{1}{10^3}+\frac{1}{10^4}+...+\frac{1}{10^{11}}\right)\)
\(\Rightarrow\frac{10N}{90}-\frac{N}{90}=\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+...+\frac{1}{10^{10}}-\frac{1}{10^2}-\frac{1}{10^3}-\frac{1}{10^4}-...-\frac{1}{10^{11}}\)
\(\Rightarrow\frac{9N}{90}=\frac{1}{10}-\frac{1}{10^{11}}\)
\(\Rightarrow\frac{N}{10}=\frac{10^{10}-1}{10^{11}}\)
\(\Rightarrow N=\frac{10^{10}-1}{10^{10}}\)
