Question

so sanh 

1/3+1/3^2+1(3^3)+...+1/3^2011+1/3^2012 voi 1/2

 

Answer 1

Đặt \(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+.....+\frac{1}{3^{2012}}\)

\(\Rightarrow3A=1+\frac{1}{3}+\frac{1}{3^2}+......+\frac{1}{3^{2011}}\)

\(\Rightarrow3A-A=\left(1+\frac{1}{3}+\frac{1}{3^2}+......+\frac{1}{3^{2011}}\right)\)\(-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+.....+\frac{1}{3^{2012}}\right)\)

\(\Rightarrow2A=1-\frac{1}{3^{2012}}\)

\(\Rightarrow A=\frac{1-\frac{1}{3^{2012}}}{2}\)

Vì \(1-\frac{1}{3^{2012}}< 1\Rightarrow A< \frac{1}{2}\)