So sánh S với 2, biết \(S = \frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + \ldots + \frac{{2023}}{{{2^{2023}}}}\).
Nhân hai vế của S với 2 để rút gọn S.
\(S = \frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + \ldots + \frac{{2023}}{{{2^{2023}}}}\)
\(2S = 1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} + \ldots + \frac{{2023}}{{{2^{2022}}}}\)
\(2S - S = \left(1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} + \ldots + \frac{{2023}}{{{2^{2022}}}}\right) - \left(\frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + \ldots + \frac{{2023}}{{{2^{2023}}}}\right)\)
\(2S - S = 1 + \left(\frac{2}{2} - \frac{1}{2}\right) + \left(\frac{3}{{{2^2}}} - \frac{2}{{{2^2}}}\right) + \left(\frac{4}{{{2^3}}} - \frac{3}{{{2^3}}}\right) + \ldots + \left(\frac{{2023}}{{{2^{2022}}}} - \frac{{2022}}{{{2^{2022}}}}\right) - \frac{{2023}}{{{2^{2023}}}}\)
\(2S - S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{2022}}}} - \frac{{2023}}{{{2^{2023}}}}\)
\(S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{2022}}}} - \frac{{2023}}{{{2^{2023}}}}\)
\(2S = 2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{2021}}}} - \frac{{2023}}{{{2^{2022}}}}\)
\(2S - S = \left(2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{2021}}}} - \frac{{2023}}{{{2^{2022}}}}\right) - \left(1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{2022}}}} - \frac{{2023}}{{{2^{2023}}}}\right)\)
\(2S - S = 2 + \left(1 - 1\right) + \left(\frac{1}{2} - \frac{1}{2}\right) + \left(\frac{1}{{{2^2}}} - \frac{1}{{{2^2}}}\right) + \ldots + \left(- \frac{{2023}}{{{2^{2022}}}} -\frac{1}{{{2^{2022}}}}\right) - \frac{{2023}}{{{2^{2023}}}}\)
\(2S - S = 2 - \frac{{2024}}{{{2^{2022}}}} + \frac{{2023}}{{{2^{2023}}}}\)
\(S = 2 - \frac{{4048 - 2023}}{{{2^{2023}}}}\)
Vậy \(S < 2\).