Question

[nguyễn ngọc linh]

 Cho S = \(\dfrac{1}{2}\) + \(\dfrac{1}{2^2}\) + \(\dfrac{1}{2^3}\) + ... + \(\dfrac{1}{2^{100}}\) Chứng minh: S < 1

 Giúp mình đi mọi người

Answer 1

[abcd]

        \(S=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{100}}\)

\(\Rightarrow\dfrac{1}{2}S=\dfrac{1}{2^2}+\dfrac{1}{2^3}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{101}}\)

trừ vế ta được :

\(S-\dfrac{1}{2}S=\dfrac{1}{2}-\dfrac{1}{2^{101}}\)

\(\Rightarrow\dfrac{1}{2}S=\dfrac{1}{2}-\dfrac{1}{2^{101}}\)

\(\Rightarrow S=2\cdot\left(\dfrac{1}{2}-\dfrac{1}{2^{101}}\right)\)

\(\Rightarrow S=1-\dfrac{1}{2^{100}}< 1\left(đpcm\right)\)