Câu hỏi của Haibara Ail

Question

So sánh tổng S vs 2S=$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+.....+\frac{2007}{2^{2007}}$

I don't need you · Accepted Answer

ta có: $S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2007}{2^{2007}}$$\Rightarrow\frac{1}{2}S=\frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+...+\frac{2007}{2^{2008}}$$\Rightarrow S-\frac{1}{2}S=\frac{1}{2}+\left(\frac{2}{2^2}-\frac{1}{2^2}\right)+\left(\frac{3}{2^3}-\frac{2}{2^3}\right)+\left(\frac{4}{2^4}-\frac{3}{2^4}\right)+...+\left(\frac{2007}{2^{2007}}-\frac{2006}{2^{2007}}\right)-\frac{2007}{2^{2008}}$$\frac{1}{2}S=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}$Gọi $Q=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2007}}$$\Rightarrow\frac{1}{2}Q=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{2008}}$$\Rightarrow Q-\frac{1}{2}Q=\frac{1}{2}-\frac{1}{2^{2008}}$$\Rightarrow\frac{1}{2}Q=\frac{1}{2}-\frac{1}{2^{2008}}$$Q=\left(\frac{1}{2}-\frac{1}{2^{2008}}\right):\frac{1}{2}=1-\frac{1}{2^{2007}}$Thay Q vào S, ta có:$\frac{1}{2}S=1-\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}$$\Rightarrow S=\left(1-\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}\right):\frac{1}{2}$$S=2-\frac{1}{2^{2006}}-\frac{2007}{2^{2007}}< 2$$\Rightarrow S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2007}{2^{2007}}< 2$Câu trả lời: ta có: $S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2007}{2^{2007}}$$\Rightarrow\frac{1}{2}S=\frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+...+\frac{2007}{2^{2008}}$$\Rightarrow S-\frac{1}{2}S=\frac{1}{2}+\left(\frac{2}{2^2}-\frac{1}{2^2}\right)+\left(\frac{3}{2^3}-\frac{2}{2^3}\right)+\left(\frac{4}{2^4}-\frac{3}{2^4}\right)+...+\left(\frac{2007}{2^{2007}}-\frac{2006}{2^{2007}}\right)-\frac{2007}{2^{2008}}$$\frac{1}{2}S=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}$Gọi $Q=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2007}}$$\Rightarrow\frac{1}{2}Q=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{2008}}$$\Rightarrow Q-\frac{1}{2}Q=\frac{1}{2}-\frac{1}{2^{2008}}$$\Rightarrow\frac{1}{2}Q=\frac{1}{2}-\frac{1}{2^{2008}}$$Q=\left(\frac{1}{2}-\frac{1}{2^{2008}}\right):\frac{1}{2}=1-\frac{1}{2^{2007}}$Thay Q vào S, ta có:$\frac{1}{2}S=1-\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}$$\Rightarrow S=\left(1-\frac{1}{2^{2007}}-\frac{2007}{2^{2008}}\right):\frac{1}{2}$$S=2-\frac{1}{2^{2006}}-\frac{2007}{2^{2007}}< 2$$\Rightarrow S=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{2007}{2^{2007}}< 2$

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