Question

Chứng minh A<1/2 biết

A = 1/3 + 1/3^2 +1/3^3+.....+1/3^99

Answer 1

\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\)

\(3A=3\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)\)

\(3A=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{98}}\)

\(3A-A=\left(1+\dfrac{1}{3}+...+\dfrac{1}{3^{98}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{99}}\right)\)

\(2A=1-\dfrac{1}{3^{99}}\Rightarrow A=\dfrac{1}{2}-\dfrac{\dfrac{1}{3^{99}}}{2}< \dfrac{1}{2}\)