[TEX] S = \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + ... + \frac{2010}{2^{2010}}[/TEX]
[TEX] 2S = 1 + \frac{2}{2} + \frac{3}{2^2} + ... + \frac{2010}{2^{2009}} [/TEX]
[TEX] 2S - S = ( 1 + \frac{2}{2} + \frac{3}{2^2} + ... + \frac{2010}{2^{2009}} ) - ( \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + ... + \frac{2010}{2^{2010}} ) [/TEX]
[TEX] S = ( \frac{2}{2} - \frac{1}{2} ) + ( \frac{3}{2^2} - \frac{2}{2^2} ) + ... + ( \frac{2010}{2^{2009}} - \frac{2009}{2^{2009}} ) - \frac{2010}{2^{2010}}[/TEX]
[TEX] S = 1 + \frac{1}{2} + \frac{1}{2^2} + ... + \frac{1}{2^{2009}} - \frac{2010}{2^{2010}}[/TEX]
\Rightarrow [TEX] 2S = 2 + 1 + \frac{1}{2} + ... + \frac{1}{2^{2008}} - \frac{2010}{2^{2009}} [/TEX]
[TEX] 2S - S = ( 2 + 1 + \frac{1}{2} + ... + \frac{1}{2^{2008}} - \frac{2010}{2^{2009}} ) - ( 1 + \frac{1}{2} + \frac{1}{2^2} + ... + \frac{1}{2^{2009}} - \frac{2010}{2^{2010}} ) [/TEX]
[TEX] S = 2 - \frac{1}{2^{2009}} - \frac{2010}{2^{2009}} + \frac{2010}{2^{2010}} = 2 - ( \frac{1}{2^{2009}} + \frac{2010}{2^{2009}} - \frac{2010}{2^{2010}} )[/TEX]
Vì : [TEX] \frac{1}{2^{2009}} + \frac{2010}{2^{2009}} > \frac{2010}{2^{2010}} \Rightarrow \frac{1}{2^{2009}} + \frac{2010}{2^{2009}} - \frac{2010}{2^{2010}} > 0[/TEX]
\Rightarrow [TEX] S = 2 - ( \frac{1}{2^{2009}} + \frac{2010}{2^{2009}} - \frac{2010}{2^{2010}} ) < 2 [/TEX]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
