Question

Cho M = \(1-\dfrac{1}{2}-\dfrac{1}{2^2}-\dfrac{1}{2^3}-\dfrac{1}{2^4}-....-\dfrac{1}{2^{10}}\) . So sánh M với \(\dfrac{1}{2^{11}}\) 

Answer 1

\(M=1-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{10}}\right)\)

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

\(2N=1+\dfrac{1}{2}+...+\dfrac{1}{2^9}\)

\(\Rightarrow2N-N=1-\dfrac{1}{2^{10}}\)

\(\Rightarrow N=1-\dfrac{1}{2^{10}}\)

\(\Rightarrow M=1-\left(1-\dfrac{1}{2^{10}}\right)=\dfrac{1}{2^{10}}>\dfrac{1}{2^{11}}\)

Vậy \(M>\dfrac{1}{2^{11}}\)