Tính giá trị biểu thức: S = 1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 10/2^10

Ta có: \[S = \frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + \frac{4}{{{2^4}}} + ... + \frac{{10}}{{{2^{10}}}}\]

Suy ra \[2S = 1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} + ... + \frac{{10}}{{{2^9}}}\]

\[2S - S = \left( {1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} + ... + \frac{{10}}{{{2^9}}}} \right) - \left( {\frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + ... + \frac{{10}}{{{2^{10}}}}} \right)\]

\[S = 1 + \left( {\frac{2}{2} - \frac{1}{2}} \right) + \left( {\frac{3}{{{2^2}}} - \frac{2}{{{2^2}}}} \right) + ... + \left( {\frac{{10}}{{{2^9}}} - \frac{9}{{{2^9}}}} \right) - \frac{{10}}{{{2^{10}}}}\]

\[S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^9}}} - \frac{{10}}{{{2^{10}}}}\]

Xét \[B = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^9}}}\]

\[2B = 2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^8}}}\]

\[2B - B = \left( {2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^8}}}} \right) - \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^9}}}} \right)\]

\[B = 2 - \frac{1}{{{2^9}}}\]

Suy ra \[S = 2 - \frac{1}{{{2^9}}} - \frac{{10}}{{{2^{10}}}}.\]