Question

How would i find the sum of the infinite series: 1/3+2/9+4/27.....

Answer 1

`S_n=(1)/(3)+(2)/(9)+(4)/(27)+...\\\\a_1=(1)/(3);\ a_2=(2)/(9);\ a_3=(4)/(27)\\\\a_2:a_1=(2)/(9):(1)/(3)=(2)/(9)\cdot(3)/(1)=(2)/(3)\\\\a_3:a_2=(4)/(27):(2)/(9)=(4)/(27)\cdot(9)/(2)=(2)/(3)\\\\this\ is\ a\ geometric\ sequence\ where\ a_1=(1)/(3)\ and\ r=(2)/(3)\\\\S_n=(a_1)/(1-r)\\\\therefore\\\\S_n=((1)/(3))/(1-(2)/(3))=((1)/(3))/((1)/(3))=1\\\\Answer:\boxed{\boxed{(1)/(3)+(2)/(9)+(4)/(27)+..=1}}`