Question

Cho biểu thức A=\(1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^{2014}}\)Hãy so sánh A với\(\frac{3}{2}\)

 

Answer 1

\(A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\)

\(\Rightarrow3A=3+1+\frac{1}{3}+...+\frac{1}{3^{2013}}\)

\(\Rightarrow3A-A\)=  \(\left(3+1+...+\frac{1}{3^{2013}}\right)-\left(1+\frac{1}{3}+...+\frac{1}{3^{2014}}\right)\)

\(\Rightarrow2A=3-\frac{1}{3^{2014}}\)

\(\Rightarrow A=\frac{3-\frac{1}{3^{2014}}}{2}\)

\(\Rightarrow A=\frac{3}{2}-\frac{\frac{1}{3^{2014}}}{2}< \frac{3}{2}\)

Vậy  \(A< \frac{3}{2}\)

Chúc bạn học tốt !!!