Question

So sánh

A= 1+ 1/2+1/2²+....+1/2¹⁰ và B=1/2¹¹

Answer 1

Giải:

\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{10}}\)

\(\Leftrightarrow\dfrac{1}{2}A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}...+\dfrac{1}{2^{11}}\)

\(\Leftrightarrow A-\dfrac{1}{2}A=\dfrac{2^{11}-1}{2^{11}}\)

\(\Leftrightarrow\dfrac{1}{2}A=\dfrac{2^{11}-1}{2^{11}}\)

\(\Leftrightarrow A=\dfrac{2\left(2^{11}-1\right)}{2^{11}}\)

\(\Leftrightarrow A=\dfrac{2^{12}-2}{2^{11}}\)

Vì \(2^{12}-2>1\)

\(\Leftrightarrow\dfrac{2^{12}-2}{2^{11}}>\dfrac{1}{2^{11}}\)

\(\Leftrightarrow A>B\)

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