Question

1. Cho B= (1/2) + (1/2)² + (1/3)³ + (1/2)⁴ + ... + (1/2)⁹⁸ + (1/2)⁹⁹

Chứng minh: B<1

Answer 1

B=1/2 +(1/2 )^2+(1/3 )^3+......+(1/2 )\(^{99}\)

⇒2B=1+1/2 +1/22 +......+1/298 

⇒B=2B−B=1−1/2\(^{99}\)

⇒1−1/2\(^{99}\) <1⇒B<1

Answer 2

\(2B=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{98}}\)

=> \(2B-B=\left(1+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{98}\right)\)\(-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}\right)\)

=> \(B=1-\frac{1}{2^{99}}< 1\)

Answer 3

\(B=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^3+......+\left(\frac{1}{2}\right)^{99}\)

\(\Rightarrow2B=1+\frac{1}{2}+\frac{1}{2^2}+......+\frac{1}{2^{98}}\)

\(\Rightarrow B=2B-B=1-\frac{1}{2^{99}}\)

\(\Rightarrow1-\frac{1}{2^{99}}< 1\Rightarrow B< 1\)