Question

cm 1/2- 1/2^2+1/2^3-1/2^4+1/2^5-1/2^6 <1/3

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Answer 1

\(A=\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-\dfrac{1}{2^4}+\dfrac{1}{2^5}-\dfrac{1}{2^6}\)

\(2A=1-\dfrac{1}{2}+\dfrac{1}{2^2}-\dfrac{1}{2^3}+\dfrac{1}{2^4}-\dfrac{1}{2^5}\)

\(2A+A=\left(1-\dfrac{1}{2}+\dfrac{1}{2^2}-\dfrac{1}{2^3}+\dfrac{1}{2^4}-\dfrac{1}{2^5}\right)+\left(\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-\dfrac{1}{2^4}+\dfrac{1}{2^5}-\dfrac{1}{2^6}\right)\)

\(3A=1-\dfrac{1}{2^6}\Leftrightarrow A=\dfrac{1}{3}-\dfrac{1}{3.2^6}< \dfrac{1}{3}\left(đpcm\right)\)