Question

What is the sum of this infinite geometric series?

Answer 1

`\qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ \displaystyle S=\sum\limits_(i=0)^(\infty)\ a_1\cdot r^i\implies S=\cfrac{a_1}{1-r}\quad \begin{cases} a_1=\stackrel{\textit{first term}}{(1)/(8)}\\ r=\stackrel{\textit{common ratio}}{(2)/(3)}\\ \qquad -1 < r < 1 \end{cases}`

`\displaystyle\sum_(k=0)^(\infty) ~~ \underset{a_1}{(1)/(8)}\underset{r}{\left( (2)/(3) \right)}^k\implies S=\cfrac{ ~~ (1)/(8) ~~ }{1-(2)/(3)}\implies S=\cfrac{ ~~ (1)/(8) ~~ }{(1)/(3)}\implies S=\cfrac{3}{8}`