Question

Cho A =  1/2+1/2²+1/2³+....+1/2^{100
chứng minh A < 1}

Answer 1

\(A=\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+..+\frac{1}{2^{99}}+\frac{1}{2^{100}}\)(1)

\(2.A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+..+\frac{1}{2^{99}}\)(2)

lay (2)-(1)

\(2.A-A=A=1-\frac{1}{2^{100}}< 1\Rightarrow dpcm\)