Question

S= 1/2 +(1/2)² +(1/2)³ +......+(1/2)²⁰¹⁸

^{Chứng minh S<1}

Answer 1

\(2S=1+\frac{1}{2}+\left(\frac{1}{2}\right)^2+...+\left(\frac{1}{2}\right)^{2017}\)

  \(S=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{2018}\)

\(\Rightarrow S=2S-S=1-\left(\frac{1}{2}\right)^{2018}\)

\(\Rightarrow S< 1\)( đpcm )

Answer 2

\(S=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{2018}\)

\(S=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2018}}\)

\(2S=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2017}}\)

\(2S-S=\left(1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2017}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2018}}\right)\)

\(S=1-\frac{1}{2^{2018}}< 1\) ( đpcm ) 

Chúc bạn học tốt ~