Đặt :
\(A=1+\frac{1}{2}+\frac{1}{4}+..........+\frac{1}{1024}\)
\(\Leftrightarrow A=1+\frac{1}{2}+\frac{1}{2^2}+..........+\frac{1}{2^{10}}\)
\(\Leftrightarrow2A=2+1+\frac{1}{2}+......+\frac{1}{2^9}\)
\(\Leftrightarrow2A-A=\left(2+1+\frac{1}{2}+....+\frac{1}{2^9}\right)-\left(1+\frac{1}{2}+....+\frac{1}{2^{10}}\right)\)
\(\Leftrightarrow A=2-\frac{1}{2^{10}}\)
Gọi dãy số trên là A
\(A=1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{1024}\)
\(2A=2+1+\frac{1}{2}+...+\frac{1}{512}\)
\(2A-A=\left(2+1+\frac{1}{2}+...+\frac{1}{512}\right)-\left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{512}+\frac{1}{1024}\right)\)
\(A=2-\frac{1}{1024}\)
\(A=\frac{2048}{1024}-\frac{1}{1024}\)
\(A=\frac{2047}{1024}\)
