Ta có :
\(A=3+3^2+3^3+........+3^{29}\)
\(\Rightarrow3A=3^2+3^3+...............+3^{29}+3^{30}\)
\(\Rightarrow3A-A=\left(3^2+3^3+........+3^{30}\right)-\left(3+3^3+................+3^{29}\right)\)
\(\Rightarrow2A=3^{30}-3\)
\(\Rightarrow A=\dfrac{3^{30}-3}{2}\)
Lại có :
\(B=\dfrac{1}{3}+\dfrac{1}{3^2}+................+\dfrac{1}{3^{29}}\)
\(\Rightarrow3B=1+\dfrac{1}{3}+\dfrac{1}{3^2}+.............+\dfrac{1}{3^{28}}\)
\(\Rightarrow3B-B=\left(1+\dfrac{1}{3}+\dfrac{1}{3^2}+.......+\dfrac{1}{3^{28}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+..........+\dfrac{1}{3^{29}}\right)\)
\(\Rightarrow2B=1-\dfrac{1}{3^{29}}\)
\(\Rightarrow B=\dfrac{1-\dfrac{1}{3^{29}}}{2}\)
\(\dfrac{\Rightarrow A}{B}=\dfrac{\dfrac{3^{30}-3}{2}}{\dfrac{1-\dfrac{1}{3^{29}}}{2}}\)
\(B=\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{29}}\)
\(3^{30}.B=3^{29}+3^{28}+...+3=A\)
\(\dfrac{A}{B}=\dfrac{3^{30}.B}{B}=3^{30}\)
