Question

Chứng minh

\(\frac{1}{2^{2+}}+\frac{1}{2^3}+\frac{1}{2^4}+........+\frac{1}{2^n}<1\)

Answer 1

Đặt   \(A=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^n}\)

=> \(2A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{n-1}}\)

=> \(2A-A=\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{n-1}}\right)-\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^n}\right)\)

=> \(A=\frac{1}{2}-\frac{1}{2^n}\) 

=> \(A=\frac{2^n}{2^n.2}-\frac{2}{2^n.2}=\frac{2^n-2}{2^n.2}=\frac{2^n+\left(-2\right)}{2^n+2^n}\)

Vì  -2 < 2ⁿ => \(2^n+\left(-2\right)<2^n+2^n\)  =>  \(\frac{2^n+\left(-2\right)}{2^n+2^n}<1\)  =>  \(A<1\)       

=> \(\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^n}<1\)   (đpcm)