Question

Cho A=1+\(\dfrac{1}{2}\)+\(\dfrac{1}{2^2}\)+\(\dfrac{1}{2^3}\)+...+\(\dfrac{1}{2^{2013}}\). So sánh A và 2

Answer 1

\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\)

\(2A=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2012}}\)

\(2A-A=\left(2+1+\dfrac{1}{2}+...+\dfrac{1}{2^{2012}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\right)\)\(A=2-\dfrac{1}{2^{2013}}< 2\)

Answer 2

2A=2+1+1/2+1/2^2+...+1/2^2012

2A-A=(2+1+1/2+1/2^2+...+1/2^2012)-(1+1/2+1/2^2+1/2^3+...+1/2^2013)

A= 2- 2^2013 < 2

Vậy A < 2