Ta có:
S = 3 + \(\dfrac{3}{2}\) + \(\dfrac{3}{2^2}\) + ........+ \(\dfrac{3}{2^9}\)
2S = 6 + 3 + \(\dfrac{3}{2^2}\) + ....... + \(\dfrac{3}{2^8}\)
Mà S = 3 + \(\dfrac{3}{2}\) + \(\dfrac{3}{2^2}\) + ........+ \(\dfrac{3}{2^9}\)
=> 2S - S = 6 - \(\dfrac{3}{2^9}\)
S = \(\dfrac{3069}{512}\)\(S=3\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^9}\right)\)
Đặt : \(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^9}\\ \Rightarrow2.A=2+1+....+\dfrac{1}{2^8}\\ \Rightarrow A=2-\dfrac{1}{2^9}\\ \Rightarrow S=3\left(2-\dfrac{1}{2^9}\right)\)
Ta có :
\(S=3+\dfrac{3}{2}+\dfrac{3}{2^2}+.........+\dfrac{3}{2^9}\)
\(2S=6+3+\dfrac{3}{2}+\dfrac{3}{2^2}+..............+\dfrac{3}{2^8}\)
\(2S-S=\left(6+3+\dfrac{3}{2}+..........+\dfrac{3}{2^8}\right)-\left(3+\dfrac{3}{2}+\dfrac{3}{2^2}+............+\dfrac{3}{2^9}\right)\)
\(S=6-\dfrac{3}{2^9}\)
\(S=\dfrac{6.512}{512}-\dfrac{3}{512}\)
\(S=\dfrac{3069}{512}\)