Question

Chứng minh 

B= 1/2+1/2²+1/2³+...+1/2^20<1

Answer 1

2B=1+1/2+1/2^2+..+1/2^19

2B-B=(1+1/2+1/2^2+..+1/2^19) -(1/2+1/2^2+1/2^3..+1/2^20)

B=1-1/2^20 <1

Answer 2

\(B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{20}}< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+..+\frac{1}{19.20}\)

\(B< \left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{19}-\frac{1}{20}\right)\)

\(B< \left(\frac{1}{2}-\frac{1}{20}\right)\)

\(B< \frac{9}{20}< 1\)

\(\Rightarrowđpcm\)

Answer 3

\(B=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\)

\(\Rightarrow2B=2.\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\right)\)

\(\Rightarrow2B=1+\frac{1}{2^1}+\frac{1}{2^2}+...+\frac{1}{2^{19}}\)

\(\Rightarrow2B-B=\left(1+\frac{1}{2^1}+\frac{1}{2^2}+...+\frac{1}{2^{19}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{20}}\right)\)

\(\Rightarrow B=1+\frac{1}{2^1}+\frac{1}{2^2}+...+\frac{1}{2^{19}}-\frac{1}{2}-\frac{1}{2^2}-\frac{1}{2^3}-...-\frac{1}{2^{20}}\)

\(\Rightarrow B=1-\frac{1}{2^{20}}\)\(< 1\)

Vậy \(B< 1\)