Question

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

so sánh A với \(\frac{3}{2}\)

Answer 1

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

\(3A=3+1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2013}}\)

\(3A-A=\left(3+1+\frac{1}{3}+...+\frac{1}{3^{2013}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\right)\)

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

\(A=\left(3-\frac{1}{3^{2014}}\right):2\)

\(A=\frac{3}{2}-\frac{1}{2.3^{2014}}<\frac{3}{2}\)

\(\Rightarrow A<\frac{3}{2}\)