Question

Find: $\frac{2}{3}+\frac{3}{3^2}+\frac{2}{3^3}+\frac{3}{3^4}+\frac{2}{3^5}+\frac{3}{3^6}+....$ to infinity.

Answer 1

Given series can be written as

$(\frac{2}{3}+\frac{1}{3})+(\frac{2}{3^3}+\frac{1}{3^3})+(\frac{2}{3^5}+\frac{1}{3^5})$ + . . . . . . . . . . . . . . . to infinity

$=1+\left(\frac{1}{3^2}\right)+\left(\frac{1}{3^4}\right)$ + . . . . . . . . . . . . . . . to infinity

Infinite G.P. with common ratio $r=\frac{1}{3^2}<1$

If $n=\infty$ and $\left|r\right|<1$ then formula for total sum is

$S=\frac{a}{1-r}$

$\therefore S=\frac{1}{1-\left(\frac{1}{9}\right)}=\frac{9}{8}=1\frac{1}{8}$