So sánh S với 2, biết S = 1/2 + 2/2^2 + 3/2^3 + ldots + — Không quảng cáo

Phương pháp giải

Nhân hai vế của S với 2 để rút gọn S.

$S = \frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} +  \ldots  + \frac{{2023}}{{{2^{2023}}}}$

$2S = 1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} +  \ldots  + \frac{{2023}}{{{2^{2022}}}}$

$2S - S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} +  \ldots  + \frac{1}{{{2^{2022}}}} - \frac{{2023}}{{{2^{2023}}}}$

$S = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} +  \ldots  + \frac{1}{{{2^{2022}}}} - \frac{{2023}}{{{2^{2023}}}}$

$2S = 2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} +  \ldots  + \frac{1}{{{2^{2021}}}} - \frac{{2023}}{{{2^{2022}}}}$

$2S - S = 2 - \frac{{2024}}{{{2^{2022}}}} + \frac{{2023}}{{{2^{2023}}}}$

$S = 2 - \frac{{4048 - 2023}}{{{2^{2023}}}}$

Vậy $S < 2$.