Ta có : \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+.....+\frac{1}{2^n}\)
\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{2^2}+.....+\frac{1}{2^{n-1}}\)
\(\Rightarrow2A-A=1-\frac{1}{2^n}\)
\(\Rightarrow A=1-\frac{1}{2^n}\)
\(\Rightarrow A=\frac{2^n}{2^n}-\frac{1}{2^n}=\frac{2^n-1}{2^n}\)