Đáp án:
$\begin{array}{l}
S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... + \frac{1}{{1024}}\\
S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{10}}}}\\
\Rightarrow 2S = 2\left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{10}}}}} \right)\\
\Rightarrow 2S = 2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^9}}}\\
\Rightarrow 2S - S = \left( {2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^9}}}} \right)\\
- \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{10}}}}} \right)\\
\Rightarrow S = 2 - \frac{1}{{{2^{10}}}} = \frac{{{2^{11}} - 1}}{{{2^{10}}}}
\end{array}$
