Question

Find the sum of the infinite geometric series (2/3)-(4/9)+(8/27)-(16/81)+...

Answer 1

`(2)/(3)-(4)/(9)+(8)/(27)-(16)/(81)+...\\\\a_1=(2)/(3);\ a_2=-(4)/(9)\\\\r=a_2:a_1\to r=-(4)/(9):(2)/(3)=-(4)/(9)\cdot(3)/(2)=-(2)/(3)\\\\|r|=\left|-(2)/(3)\right|=(2)/(3) \ \textless \ 1\\\\therefore\\\\S=(a_1)/(1-r)\\\\subtitute\\\\S=((2)/(3))/(1-\left(-(2)/(3)\right))=((2)/(3))/(1+(2)/(3))=((2)/(3))/((5)/(3))=(2)/(3)\cdot(3)/(5)=\boxed{(2)/(5)}`