Question

Cho biểu thức A=1+1/3+1/3^2 +...+1/3^2014 hãy  so sánh  A với 3/2

Answer 1

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

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

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

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

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

Answer 2

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

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

\(\Rightarrow3A-A=3-\frac{1}{3^{2014}}\)

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