Question

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

Answer 1

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