Câu 18 (0,5 điểm): Cho 2 biểu thức: A= $\frac{1}{2}$+ $\frac{1}{2^2}$+ $\frac{1}{2^3}$+ $\frac{1}{2^4}$+...+ $\frac{1}{2^{2021}}$+ $\frac{1}{2^{2022}}$  và B= $\frac{1}{3}$+ $\frac{1}{4}$+ $\frac{1}{5}$+ $\frac{17}{60}$  So sánh A và B?

 Ta có

A=$\dfrac{1}{2^1}$+$\dfrac{1}{2^2}$+$\dfrac{1}{2^3}$+$\dfrac{1}{2^4}$+...+$\dfrac{1}{2^2021}$+$\dfrac{1}{2^2022}$

2A=1+$\dfrac{1}{2^1}$+$\dfrac{1}{2^2}$+$\dfrac{1}{2^3}$+...+$\dfrac{1}{2^2020}$+$\dfrac{1}{2^2021}$

2A-A=1+$\dfrac{1}{2^1}$+$\dfrac{1}{2^2}$+$\dfrac{1}{2^3}$+...+$\dfrac{1}{2^2020}$+$\dfrac{1}{2^2021}$ -$\dfrac{1}{2^1}$+$\dfrac{1}{2^2}$+$\dfrac{1}{2^3}$+$\dfrac{1}{2^4}$+...+$\dfrac{1}{2^2021}$+$\dfrac{1}{2^2022}$ 

A=1-$\dfrac{1}{2^2022}$

 => A<1

Ta lại có

B=$\dfrac{1}{3}$+$\dfrac{1}{4}$+$\dfrac{1}{5}$+$\dfrac{17}{60}$

B=$\dfrac{20}{60}$+$\dfrac{15}{60}$+$\dfrac{12}{60}$+$\dfrac{17}{60}$    

B=$\dfrac{20+15+12+17}{60}$=$\dfrac{64}{60}$>1

=> A<1<B => A<B

 Vậy A<B

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