Question

Cho S=\(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{30}}\)

Chứng tỏ: S<1

Answer 1

2S = 1 + 1/2 + 1/2^2 + ... + 1/2^29

2S - S = 1- 1/2^29

S =  1 - 1/2^29 < 1

vậy S < 1

Answer 2

S=\(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{30}}\)

2S= \(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{29}}\)

2S - S=( \(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{29}}\)) - (\(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{30}}\))

S= \(1-\frac{1}{2^{30}}\)

S= \(\frac{2^{30}}{2^{30}}-\frac{1}{2^{30}}\)

S= \(\frac{2^{30}-1}{2^{30}}\)