Đặt: \(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2048}\)
\(\Rightarrow A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{11}}\)
\(\Rightarrow2A=2\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{11}}\right)\)
\(=1+\frac{1}{2}+\frac{1}{2^2}+.....+\frac{1}{2^{10}}\)
\(\Rightarrow2A-A=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{10}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{11}}\right)\)
\(\Rightarrow A=1-\frac{1}{2^{11}}=\frac{2^{11}-1}{2^{11}}=\frac{2047}{2048}\)
\(2A=1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{1024}\)
\(2A-A=\left(1+...+\frac{1}{1024}\right)-\left(\frac{1}{2}+...+\frac{1}{2048}\right)\)
\(A=1-\frac{1}{2048}=\frac{2047}{2048}\)
\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+....+\frac{1}{2048}\)
\(=\left(1+\frac{1}{2}+\frac{1}{4}+....+\frac{1}{2048}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2048}\right)\)
\(=1-\frac{1}{2048}\)
\(=\frac{2047}{2048}\)
Kết quả là 2047 / 2048 .
Có cần giải chi tiết không ?
( 1 + 1/2 - 1/2 + 1/4 - 1/4 + 1/8 - 1/8 + 1/16 - 1/16 + .....- 1/2048)
= 1 - 1/2048
= 2047/2048
